Arsenal had United at arm's length for the remainder of the contest, seeing out the game with professionalism and giving their manager a victory to savour in a forgettable season.
Granit Xhaka and former United player Danny Welbeck scored goals in a four-minute span early in the second half to end United's 25-match unbeaten streak in the league dating to October.
Wenger insisted his team still had an outside chance of qualifying for next season's Champions League.
The ex-United striker said: "It's really important to get the win".
Proceedings had long since lost any real tempo, though, and Arsenal comfortably held on to move to within two points of United and six of Manchester City, while the Gunners still have a game in hand on both.
United were much-changed against Arsenal with a Europa League semi-final second leg against Celta Vigo on Thursday, but how does Neville think this loss will affect his former club?
The 19-year-old made his Premier League debut at The Emirates but put in a strong display at right-back to help keep Alexis Sanchez quiet in north London.
With Darmian shadowing Mesut Ozil on the other half of the pitch, Mkhitaryan and Juan Mata - back after a seven-week lay-off with a groin injury - found themselves playing as auxiliary wing-backs.
Aaron Ramsey and Alex Oxlade-Chamberlain forced fine saves from David De Gea in the first half, while Wenger's side twice played the same type of risky ball across the United box that only needed a different bounce to go in.
It means their chances of sneaking into the top four would appear increasingly remote.
Welbeck saw a couple of close-range shots blocked before Alex Oxlade-Chamberlain's deflected effort was pushed away by De Gea.
A lapse in concentration from Rob Holding nearly gifted United a goal just beyond the half-hour.
The Reds remained in fifth following the result at the Emirates as two second-half goals from the Gunners proved to be the difference between the sides.
But after 54 minutes the home side broke through when Granit Xhaka picked up possession 30 yards out and his shot took a big deflection off Herrera and looped over De Gea.
Before United had had time to clear their heads, it was 2-0.
Welbeck made it 2-0 three minutes later, heading in off the bar from Oxlade-Chamberlain's cross.
Mourinho attempted to claw United back into the contest by introducing Jesse Lingard and Marcus Rashford form the bench, but neither had a huge impact.
Mourinho gave a debut to 20-year-old midfielder Scott McTominay in the closing stages, but there was little else for United to cherish.