Arsenal unbeaten run ended by Tarkowski who nodded in to earn Everton an important win
Sean Dyche's reign as Everton manager began in the best possible way, with a 1-0 win over Premier League leaders Arsenal lifting his new side out of the relegation zone.
Inside a fired-up Goodison Park, Everton were full of intensity and their set-piece winner will have delighted Dyche, with James Tarkowski out-muscling Gunners captain Martin Odegaard at the back post to head in from a corner.
Everton could easily have been ahead before then, with several good chances falling Dominic Calvert-Lewin's way in the opening 45 - although Conor Coady had to come up with a goal-line clearance to deny Bukayo Saka late in that half.
Arsenal began the second period with purpose but never got up to the levels they showed at home to Manchester United last time out in the Premier League and could never seriously trouble a resolute Everton backline in the final stages.
They will now have a nervous eye on Super Sunday where Manchester City can close their gap at the top of the table to just two points with a win at Tottenham.
At the other end, the standings suddenly look a lot more encouraging for Everton ahead of Saturday's other games, with their first win in nine league matches moving them to 17th, a point clear of the drop zone. Their fans had protested against the board before the game but left Goodison celebrating a vital three points.