Everton were trailing 1-0 to Antonio Valencia’s early stunner but playing well and threatening an equaliser.
Then Ashley Williams gifted United a second goal scored by Henrikh Mkhitaryan and the floor gave way on Everton.
To rub salt into the wounds, Romelu Lukaku, who joined United from Everton for 75 million pounds ($101.85 million) in July, made it 3-0, appearing to enjoy kicking his old side when they were down.
Anthony Martial’s injury-time penalty completed the rout, meaning Everton have conceded 10 goals without scoring in eight days. They have lost their last four games and have not won since beating Hajduk Split in the Europa League on Aug. 17.
Few would have expected Koeman’s name to feature highly on the “managers at risk” list after the club’s spending spree in the close season which saw players such as Gylfi Sigurdsson, Davy Klaassen and Michael Keane arrive for big fees.
Now, Wednesday’s League Cup clash at home to second-tier Sunderland takes on huge importance with bookmakers William Hill making Koeman second favourite to be the next manager sacked.
“After today we have four home games and we have to win, otherwise we’ll have problems,” the Dutchman said.
Everton, who in mitigation have played Manchester City, Chelsea, Tottenham Hotspur and Manchester United already in the league, slipped below West Ham on goal difference to 18th spot. Their next league games are at home to Bournemouth and Burnley.
Koeman said his side were “afraid” and lacking confidence in the early stages at Old Trafford but that he was encouraged by the way they responded, especially after a dreadful Europa League display away in Italy to Atalanta on Thursday.
“We grew in the game and in confidence and not many teams will get two big chances like we got today. We didn’t score the goal and after a personal mistake it was 2-0,” he said.
“From what I saw today from the team it made me happy. More happy than Thursday.”
Everton midfielder Morgan Schneiderlin, who conceded the stoppage-time penalty, said had Everton taken their chances, most notably two good ones for Wayne Rooney, it could have been a very different outcome.
”If we had pulled back to 1-1 it would have been a different game,“ he said. ”We have to take the positives. Our show on Thursday was horrendous. It was a case today that we were fighting for the shirt.
“No one wants to lose 4-0 but we found our aggression again.”