Antonio Valencia's piledriver volley - a goal of the season contender - was all that separated the sides as the clock ticked into the last ten minutes.
Even though United's lead had rarely come under threat, there was a feeling of relief around Old Trafford when Henrikh Mkhitaryan swept home the match-sealing second on 82 minutes.
Romelu Lukaku (who else) then made it 3-0 before substitute Anthony Martial added further gloss with a late penalty.
The result saw United go joint top with rivals City - who had won 6-0 at Watford on Saturday - and the two sides boast identical records, with the same number of points, goals scored, conceded and the same goal difference.
United were on the front foot from the off and got off the perfect start when we took the lead after only four minutes, in swashbuckling style.
Man of the match Nemanja Matic picked out Valencia, and he smashed home a spectacular 25-yard volley beyond a helpless Jordan Pickford - his first Old Trafford goal since January 2014.
The Reds continued to press and could have been out of sight against opponents bereft of confidence and any semblance of form.
Juan Mata collected a stray pass from Michael Keane and played in Lukaku but, with the goal at his mercy, the red-hot striker could only pull his effort wide.
Mata himself went close, as did Rooney at the other end, before David de Gea saved well from Tom Davies, who was offside anyway.
After the break, the visitors began to improve and carved out two good chances in quick succession.
De Gea denied the returning Rooney with a smart close-range block, and the keeper then also saved well from Everton's £45m summer signing Gylfi Sigurdsson three minutes later.
Mata struck a post from a free-kick, but the result was soon put beyond any lingering doubts when Mkhitaryan stroked home his first goal of the season having been played in unselfishly by Lukaku.
The in-form Belgian got in on the act himself with a minute to play.
His free-kick hit the wall but he remained alive to the loose ball and tucked in past Pickford after substitute Jesse Lingard had flicked on a deflected Matic cross.
There was still time for another sub, this time Martial, to add a flattering sheen to the scoreline when he slotted home from the spot after ex-Red Morgan Schneiderlin had handled.
De Gea preserved another clean sheet - his 100th for the club - with a brilliant late save to keep out Sandro Ramirez's long-range effort, on another memorable afternoon at Old Trafford.
Overall team performance: 7/10
United Faithful Man of the Match: Nemanja Matic. Immense again.