Everton took the lead from Bernard’s shot, but Bruno Fernandez’s two goals put United ahead at half-time.
Edinson Cavani’s first goal for United wrapped up the win in stoppage time to provide some much-needed relief to manager Ole Gunnar Slusskaire.
It was the visitors’ third consecutive win in a single season in the Premier League, having lost three times to Salisbury and saw a draw at home.
The victory pushed United to 13th in the Premier League, six points behind leaders Southampton.
The United’s Norwegian manager got into an altercation in a post-match televised interview, complaining of early play on Saturday afternoon, saying the game was played on Wednesday against Basaksir in Istanbul, Turkey.
“We were set up to fail, Salisbury told BT Sport.” Kickoff time set us up to fail. We went to Turkey, already played a lot of games this season, we came back on Thursday morning and we are playing Saturday lunch, it’s a shame.
“The authorities set us up to fail, who is responsible? We had enough of it. The players are venting this season, an epidemic, mentally and physically. Let’s play on Sunday, from having an international break. After all, it’s a joke.
“The problem was that when we scored, we didn’t defend well, and they took advantage,” Ancelotti told BT Sport. “They were able to score easily. We said before the game that we have to defend better. We were slow at the back, we have to improve.
“I hope that after the international break we will come back in a different form. It’s not just the defenders’ problem, it’s the whole team.”
Everton forward Richardson was sent off in a draw against Liverpool and was later banned for three matches and Everton lost the talented Brazilian.
Following the upcoming international break, United play West Bromwich Albion on November 21, while Everton are out of Fulham.