Everton s English defender James Tarkowski celebrates after scoring a goal during the English Premier League football match between Everton and Arsenal at Goodison Park in Liverpool, north-west England, on February 4, 2023. AFP
Everton's first win in 11 games lifts the Toffees out of the relegation zone.
Arsenal remain five points clear of Manchester City at the top of the table, but may rue a huge chance to extend their advantage as they aim for a first league title since 2004.
Everton came into the game on a run of four league defeats at Goodison for the first time since 1958.
That disastrous form cost Frank Lampard his job and Dyche's appointment got the desired bounce in performance.
Everton's board again stayed away due to rising tensions between the club's hierarchy and supporters.
Protests were planned for before and after the game for fans to voice their anger at the running of the club.
The Toffees were the only Premier League club to not make any signings in the January transfer window despite their perilous position in the table.
But the home side showed there is still enough talent available to Dyche to dig themselves out of trouble if they can harness the same level of energy and intensity on a more regular basis.
Two of Dyche's former Burnley players were included in his first starting line-up and they combined for the only goal of the game on the hour mark as Tarkwoski headed in Dwight McNeil's corner.
A first victory since October was no more than Everton deserved after they passed up a host of chances to go ahead before the break.
Keeping Dominic Calvert-Lewin fit will be critical to Everton's survival hopes, but the England international showed his rustiness in front of goal in a second consecutive injury-impacted season.
Amadou Onana's surging run and ball across the face of goal just needed a finishing touch that Calvert-Lewin could not stretch to make.
Moments later Abdoulaye Doucoure, who had been frozen out by Lampard, headed a glorious chance wide when unmarked inside the box.
It took Arsenal half an hour to even threaten Jordan Pickford's goal, but they should have led when Eddie Nketiah fired high and wide with just the goalkeeper to beat.
Conor Coady then had to react quickly to turn Bukayo Saka's volley off the line as the visitors began to find their rhythm.
But it was Calvert-Lewin who passed up another sight of goal in first half stoppage time when he headed Seamus Coleman's cross wide.
Arsenal were brighter at the start of the second period and Martin Odegaard will feel he should have hit the target when teed up by Nketiah.
But just as the Gunners were beginning to turn the screw, their luck at the back ran out.
Tarkowski was fundamental to Dyche overachievements in keeping Burnley in the Premier League for six consecutive seasons between 2016 and 2022 and the centre-back rose highest to power home McNeil's corner on the hour mark.
Mikel Arteta's decision to substitute three of his most consistent performers this season in Gabriel Martinelli, Thomas Partey and Odegaard will be questioned as Arsenal lacked the guile to break down the mass ranks of defence from Dyche's men.
The full-time whistle was met with jubilant celebrations as Everton lifted themselves one point clear of the bottom three.
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