James Tarkowski's second-half header was enough for Everton to seal a crucial win in Sean Dyche's first game as the club's new manager.
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Mikel Arteta's side missed out on the chance to go eight points clear of second-placed Manchester City after losing only their second league game of the campaign.
The result also sees Everton climb out of the bottom three as they look to avoid being relegated from the Premier League for the first time in their history.
Tarkowski, who Dyche signed when he was at Burnley, said he hoped the win could help change the prospects for Everton, who had only won three league games all season before Saturday.
"It has been tough for us and the fans and for all the staff in the club. Hopefully this is the turning point for the club, there is a lot of talent here," he told BT Sport.
"It's taken me a while, I need to score more. What a great time to do it and a great day. It's back to work on Monday and we crack on.
"We've lacked that little bit of grit at times and that's what the fans want. We've seen what it's all about today."
Everton started the game brightly and made it difficult for Arsenal to find their rhythm early on.
However, the league leaders had the first big chance after 30 minutes when Eddie Nketiah blasted over from close range.
- Arsenal fail physical test as Dyche makes instant impact
The game came to life after that as Calvert-Lewin came agonisingly close to tapping home from Amadou Onana's cross while Abdoulaye Doucoure's close-range header went wide.
Moments later, Bukayo Saka saw his fine effort blocked off the line from Conor Coady and minutes before the break, Calvert-Lewin's looping header missed the target.
Everton continued their fine form in the second half and took the lead after 59 minutes when Tarkowski headed home from Dwight McNeil's corner.
Arsenal responded to the goal with Leandro Trossard forcing Jordan Pickford into a good save while Saka blazed over from inside the box.
Oleksandr Zinchenko had a decent chance to equalise late on but his shot went over the bar.
The hosts managed to hold on to claim their first victory in the league since October.
Everton winger McNeil added: "We know how good they [Arsenal] are as a team, but we also know how good we are as a team as well with all the chances we had."
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Everton vs Arsenal result, score, highlights and analysis as James Tarkowski stuns Premier League leaders
James Tarkowski's second-half header gave Everton victory at home over Arsenal and moved them out of the Premier League relegation zone in new head coach Sean Dyche's first game in charge.
Centre-back Tarkowski found space among a crowd of players to nod the winner from close range and reward a much-improved performance from the Toffees, who had lost their previous four Premier League home games.
Arsenal arrived with one defeat in this season's top flight but struggled to cope with Everton's pressing for long spells of the game as they slipped to a third defeat in as many visits to Goodison Park.
Bukayo Saka saw his volley from a corner cleared off the line by Conor Coady during the first half and goalkeeper Jordan Pickford also denied substitute Leandro Trossard as Arsenal pushed in vain for an equaliser to strengthen their hopes of sustaining their advantage over second-placed Manchester City.
Revitalised-looking Everton deserved their win, almost taking the lead when Dominic Calvert-Lewin and Amadou Onana failed to connect effectively with crosses before Tarkowski found a more decisive response to a delivery from Dwight McNeil, who had been part of Dyche's squad at Burnley alongside the matchwinner.
The result moved Everton up to 17th before the rest of the Premier League action on Saturday, a point clear of Wolves and Bournemouth.
City can close the gap on Arsenal when they visit Tottenham Hotspur on Sunday, although the pace-setters will have a game in hand on the reigning champions after that match.
Dyche draws best from Everton
Having been appointed to a crescendo of indifference at the start of the week, Dyche had wisely spoken about the need to drown out what he calls the “outside noise” surrounding Everton.
Dyche defended Everton’s perceived lack of effort to bring in new faces before transfer deadline day on Tuesday, when they sold Anthony Gordon to Newcastle United for £45 ($49.5m), but is a wily enough manager to know that only results can improve the mood around the club.
The pragmatic 51-year-old even encouraged his players to provide anonymous feedback on their time at the club during his first week on the training ground.
During the opening minutes of this game, a plane was seen overhead the stadium appearing to deride Everton as the worst-run club in the Premier League.
On the pitch, though, they looked transformed in pursuit of their first win since October, troubling Arsenal with their energy from the off.
Calvert-Lewin would have scored had he been an inch further forward to meet an excellent early delivery from Onana, who then turned villain by not appearing to realise how much space he had with a free header in front of goal.
Those misses mattered far less when Tarkowski, making his 200th Premier League start, found a slither of space to give his side a warranted advantage.
Everton's first win in 10 games had not been expected to come against opponents who have flown through a multitude of tricky games this season. Their start under Dyche - a man who was second choice to Marcelo Bielsa - bodes highly promisingly for a more stable second half of the season.
More to follow
The Sporting News followed the match live and provided live score updates and commentary.
Full-time: Everton 1-0 Arsenal
It's fair to say there are scenes around Goodison Park! Sean Dyche has the desired effect on Everton, producing a shock against the leaders to move Everton out of the relegation zone and leave the fans clearly ecstatic here at Goodison Park.
Thanks for following our live coverage! We'll have a report and analysis at the top of this page shortly.
90+6 mins: Everton 1-0 Arsenal
Another large helping of fisticuffs on the touchline, with Aaron Ramsdale racing in to get involved after Oleksandr Zinchenko and Neal Maupay clash near the halfway line.
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The always-innocent Maupay ends up lying on the floor on his back and appearing to grin as hostilities take place around him. All rather silly, in the context of a game that hasn't been particularly ill-tempered.
Maupay is eventually booked.
90 mins: Everton 1-0 Arsenal
Arsenal have wasted two half-decent opportunities. The home crowd around Goodison Park are waving their fists, grinning and generally looking rather pleased indeed.
86 mins: Everton 1-0 Arsenal
Jordan Pickford is booked for timewasting, picking up his fourth caution of the campaign.
Naturally, he shrugs his shoulders and asks the referee what on earth he can mean by suspecting him of stalling for time against the league leaders with a one-goal advantage and about five minutes to go.
85 mins: Everton 1-0 Arsenal
Mikel Arteta swaps Ben White for Takehiro Tomiyasu, who's clapping and cajoling his teammates even before he's crossed the line.
82 mins: Everton 1-0 Arsenal
Aaron Ramsdale bowls the ball out urgently every time he gets it now. That five-point lead over Manchester City is going to look very slender if Arsenal lose this game; they've got City in two games' time.
79 mins: Everton 1-0 Arsenal
Mikel Arteta shows his annoyance on the touchline after Bukayo Saka lumps a shot over the crossbar from a tight angle inside the penalty area on the right.
78 mins: Everton 1-0 Arsenal
Leandro Trossard turns inside the penalty area on the left and crashes in a swerving shot.
It's straight at Jordan Pickard but the venom on the shot means the England goalkeeper has to beat it away with two strong arms. Well played.
75 mins: Everton 1-0 Arsenal
Pinball in the Arsenal goalmouth following an Everton corner from their left. Neal Maupay flashes a shot well over the crossbar on the turn.
72 mins: Everton 1-0 Arsenal
James Tarkowski is making his 200th Premier League start today. Four of his eight goals in the top flight have been headers from corners.
69 mins: Everton 1-0 Arsenal
Amadou Onana receives his sixth booking of the season for a nasty challenge on Ben White by one of the touchlines.
Then Neal Maupay engages in some pushing and shoving with Arsenal players, as he once did for former side Brighton & Hove Albion. It's all getting somewhat tetchy.
65 mins: Everton 1-0 Arsenal
Dyche is back on more animated form now after being temporarily gobsmacked by that goal, perhaps.
He's reeling his players around on an imaginary fishing rod with his arms. If ever you wanted a manager to orchestrate a successful rearguard action, Dyche is the man for the job.
62 mins: Everton 1-0 Arsenal
That goal must have been pretty much the only incident this afternoon that Sean Dyche hasn't reacted to, standing still amid the bedlam around him. Cold as Dyche, you might call him.
He's brought Neal Maupay on for Dominic Calvert-Lewin, which is one way to react to going ahead.
60 mins: GOAL – Everton 1-0 Arsenal (James Tarkowski)
Everton have the goal they deserve! The Sean Dyche effect is in full swing as Dwight McNeil swings in a corner from the right - Everton have now had five of the six corners today - and James Tarkowski finds space among an extremely tight crowd to head in from close range!
59 mins: Everton 0-0 Arsenal
Mikel Arteta makes a couple of changes, new signing Jorginho and Leandro Trossard coming on for Thomas Partey and Gabriel Martinelli.