Arsenal were winless in their last three Premier League games ahead of their trip to Merseyside and were dumped out of the Champions League mid-week at Barcelona.
However, former Manchester United striker Danny Welbeck made the breakthrough in the seventh minute after he completed an impressive passing move from the visitors.
Wenger’s men doubled their lead three minutes before half-time thanks to Alex Iwobi’s clinical finish on his Premier League debut.
Everton rarely troubled David Ospina’s goal and Arsenal wasted chances to add a third but sealed their first victory since 14 February.
Former Everton midfielder Nevin was disappointed with the performance of Roberto Martinez’s side, who were unable to test a tiring Arsenal side in the second half.
“I expected Arsenal to tire near the end of the game which is what happened, but Welbeck, Iwobi and Ozil all looked good,” Nevin told BBC Sport.
“Over 90 minutes Arsenal were the only team who looked likely to win this game.”
Wenger’s side are in action against London rivals Watford in their next Premier League game at the Emirates Stadium.
Arsenal haven’t won the Premier League title since Wenger’s famous Invincibles side stormed to title glory in 2004.
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