Romelu Lukaku scored against his former club as Manchester United thrashed Everton to move joint top of the Premier League.
Wayne Rooney's return to Old Trafford ended in a dismal defeat - embossed by three late goals - which sent Everton into the bottom three.
The Toffees have endured a dire start to the season and have not scored a goal in any of four successive defeats in all competitions.
Antonio Valencia smashed a shot past Jordan Pickford from 25 yards to put Jose Mourinho's side into an early lead which rarely came under threat.
Rooney, United's record goalscorer, had Everton's two best opportunities but once he left the pitch with nine minutes to go his former employers hit overdrive.
Lukaku found Henrikh Mkhitaryan to steer home before the Belgium striker bundled in the third himself at the far post, and substitute Anthony Martial added a stoppage-time penalty.
United and Manchester City have an identical record at the top after five Premier League games - wins, points, goals scored and goals conceded - with Pep Guardiola's team on top only alphabetically. If the two teams had identical records at the end of the season, there would be a play-off.
Lukaku, who left Everton to join United for £75m in July, was kept quiet for most of the game but set up Mkhitaryan with a smart pass to punish a costly Ashley Williams mistake, and then celebrated in front of the away fans when he scored the third a minute from time.
The Belgium international has scored five goals in his five league appearances for his new club but did miss a great chance in the first half.
Clean through on goal after another defensive error, he fired wastefully wide in front of the Stretford End.
But he took his chance clinically in the last minute. His initial free-kick hit the wall, but when the ball was played back in, he latched on to Jesse Lingard's flick to fire in, before cupping his ears to the Everton fans.
Lonely return for Rooney
Rooney scored 253 goals for United in his 13 years at the club, but after an expected and justified warm welcome on his return, he cut a frustrated and lonely figure at times.
His move back to boyhood club Everton in July was widely heralded, and his two goals this season remain the club's only strikes in the Premier League.
England and Manchester United may have decided in recent years that his best position is not up front but he started there and was isolated for long periods before moving deeper in the second half.
He had Everton's two best chances, steering a shot wide in the first half and then shooting straight at David de Gea seconds after the interval.
Rooney was given a standing ovation and handshake from Mourinho when replaced inside the last 10 minutes - after which Everton collapsed. It is now 325 minutes since they scored in the league.
Everton in trouble?
Koeman spent lavishly this summer on Rooney, Gylfi Sigurdsson, Michael Keane, Jordan Pickford and Davy Klaassen to build - on paper at least - a strong Premier League squad.
But in failing to replace Lukaku, he left the side short of pace and mobility up front - as well as goals.
The Toffees turned in a dismal performance against Tottenham last weekend and were even worse against Atalanta in their Europa League opener on Thursday night.
Sunday's trip to Old Trafford always looked an ominous way to end a tough week and from the moment Valencia cracked in his sensational opener, the writing was on the wall for a manager who had admitted before the match that he was "worried".
After the game, he admitted that his side looked "afraid" and lacked confidence.
The men in blue were perhaps blameless for Valencia's superb strike but they were so deep in the opening exchanges that Sigurdsson was practically playing at left-back.
Everton have faced four of last season's top six in their opening five games, but have only mustered 10 shots on target all season.
They improved markedly in the second half - before United's late flurry of goals - but are yet to get the best out of their new signings. and find themselves in the relegation zone for the first time since the opening day of the 2011-12 season.
Koeman now has four games in a row at home. He needs wins.
'We deserved more'
Everton manager Ronald Koeman: "We deserved more than we got. We made a big mistake by Ashley Williams at 1-0 and after that it was over.
"What I saw from the team made me happy. More happy than Thursday at Atalanta.
"It's really difficult, as a manager you try to help and support them - but when the whistle starts it's all about them. We started too afraid.
"We have four home games in different competitions and we need to win, otherwise we'll have problems."
Manchester United boss Jose Mourinho: "The first 30-35 minutes was probably our best performance of the season.
"Everton are going to be fine but they are living a difficult moment.
"The fixtures they got, I wouldn't like these fixtures for me all together at the same time. I would like to play against one of the top five - but all of them, that is not fair for them."
Man of the match - Nemanja Matic (Manchester United)
Can we play you every week?
■ Manchester United have now beaten Everton 34 times in the Premier League, a joint record in the competition (also held by United, who have beaten Aston Villa 34 times).
■ At his sixth attempt in the Premier League, Jose Mourinho finally defeated Ronald Koeman, ending a run of five matches without a win against the Dutchman (D4 L1).
■ Everton have scored four goals in their last 10 Premier League games - they had scored 26 goals in the 10 prior to this run.
■ The Toffees are winless in their last 11 away Premier League games (D5 L6), their worst run since May 2004 (also 11 without a win).
■ Romelu Lukaku has scored five goals in his first five Premier League appearances for Manchester United, a record matched only by Louis Saha and Robin van Persie.
■ Everton have failed to score in four consecutive matches in all competitions for the first time since April 2006.
Manchester United face Burton Albion in the EFL Cup on Wednesday before a rare Saturday 15:00 BST engagement at Southampton. Everton start that run of home games against Sunderland in the cup on Wednesday, with Bournemouth the visitors on Saturday.