Granit Xhaka opened the scoring for Arsenal in the 12th minute with a strike from outside of the 18-yard-box.
The Switzerland international’s shot – which was came from nearly 25 yards – bamboozled David de Gea, who misread the flight of the ball and was found wanting.
The error was out of character for the Spaniard, who is considered by many to be the best goalkeeper in the world.
Ex-Leeds and Manchester City defender Mills wasn’t impressed by the former Atletico man’s reading of the shot, which he reckons was poor for a keeper of his standing.
“David de Gea just went completely the wrong way,” the former England international said on BBC Radio 5 Live.
“It didn’t move that much and it has to down as a huge goalkeeping error.
“There is a little bit of swerve on the ball but it’s no Roberto Carlos is it?!”
Mills believes the Gunners should continue to test de Gea’s resolve with more shots from outside of the area.
He said: “If you’re an Arsenal player now, if you get the ball anywhere near goal you take a shot.
“It doesn’t matter how good David de Gea is, test his nerve.”
The BBC pundit later added: “It’s an incredibly poor error from David de Gea. It wasn’t a particularly well hit shot and it’s just wrong-footed him.
“Arsenal have looked good. They’ve dominated and have created a lot. Manchester United look sluggish in the final third.
“They’ve maybe got a bit of a hangover from last Wednesday and Ole Gunnar Solskjaer has got a bit team talk on his hands.
Xhaka’s strike was the sixth goal de Gea has conceded from outside of the box in the Premier League this season.
Last campaign, he only conceded three goals from outside of the area.