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.
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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.