It was always going to be a tough match for United, and they made it even tougher for themselves at times, hitting quite a few of the misplaced passes that haunted them at Old Trafford last weekend.
A typical Javier Hernandez goal helped ease the Red Devils to a 1-0 win, but it was in the away side's own final third where the difference was made.
After the harrowing performance in their last Premier League game, a match in which United not only conceded six goals but also 22 shots, it was vital for the team more than anything to keep a clean sheet.
They achieved that at Aldershot in the League Cup on Tuesday, but the real key for the road to recovery was to do that in an important Premier League match.
And the man who made that happen was the defensive rock himself, Nemanja Vidic.
His absence in the game with Manchester City was absolutely deleterious for Manchester United, as proven by Jonny Evans' costly dismissal and Rio Ferdinand's lack of understanding with his centre-back partner.
And his absence in all the other Premier League games—barring the one he did feature in against West Bromwich Albion back in August—has also affected United greatly.
It's clear to see how important the former Serbia international is for Manchester United, and it was emphasized against Everton.
United's back-line appeared its most calm this season as Vidic returned to the side, despite coming under intense pressure towards the latter stages of the match against Everton.
The strong Serb's no-nonsense approach to clearing the defensive lines was vital, as he made 10 clearances, six of them effective (i.e. cleared to a teammate), which was more than any other player on the pitch.
This allowed United the respite they needed, and have lacked too often after playing the ball around at the back, in defence before the next wave of opposition attack.
To put the importance of his clearing ability into perspective, Rio Ferdinand made the most clearances for United in that match against City.
He made five clearances, one of them effective. His team conceded six goals.
So, it's evident that Nemanja Vidic's clearing ability is paramount to the United cause.
However, besides that Vidic also brings to the table the strong tackling and aerial ability that has been lacking in the Red Devils' defence recently.
He won three aerial duels in the match against Everton, the second highest of any player on the pitch (with Phil Jones winning four), and has the best aerial duel success rate of any United defender this season.
It's that sort of ability in the air which also shows just how important he is to the team and how vital he will be in the coming matches.
The road to recovery has now begun for Manchester United, and as proved against Everton, the key man in United's quest to regain Premier League glory is Nemanja Vidic, with his efficient clearances, robust tackling, aerial strength and overall calming influence making the difference for the Red Devils.