Manchester City's bid to qualify for the Champions League for the 15th time in a row was boosted by two late goals against Everton that they hardly deserved.
The game was heading for a draw, which would have left them vulnerable to Chelsea, who face Nottingham Forest on Monday, but 20-year-old Nico O'Reilly opened the scoring in the 84th minute.
Substitute Mateo Kovacic doubled the lead in stoppage time to extend the gap to four points ahead of sixth-placed Stamford Bridge.
Yet a ninth successive – and final – victory at Goodison Park had long seemed unlikely as, despite their desire to do better than the home side, whose survival had long been, if not mathematically, assured, they lacked energy, motivation and team cohesion.
Everton goalkeeper Jordan Pickford did not fully return to the pitch until the 72nd minute, when he dived to his right to stop Omar Marmouche, who should have scored after a mistake by substitute Michael Keane, but the England international put out his hand.
Then Keane, making his first appearance since early February following James Tarkowski's hamstring injury, failed to track O'Reilly's run for the opener and from that moment City found the confidence they had been missing.
Half an hour into the match, Vitaliy Mykolenko found space on the left behind City's lax pressing and Bernardo Silva turned and threw his arms wide, challenging Matheus Nunes, 50 yards behind him. Nunes responded with a similar gesture.
Meanwhile, Everton earned a corner, from which Tarkowski headed the ball into the post.
It was the closest they came to scoring in the first three quarters of the game, despite Kevin De Bruyne being given free rein at Goodison Park, moving around as a false nine in an attempt to find some space to create.
However, it made little difference due to a lack of activity from his teammates, with Savinho failing to finish a pass from the far post and then complaining unsuccessfully about interference from Jake O'Brien.
The Everton defender certainly made a difference late in the half, diving towards De Bruyne's goal and heading wide, just moments after Jack Harrison felt he should have been given a penalty for hitting Savinho's arm.
Surprisingly, Pep Guardiola made no changes at half-time, but his confidence in his starting XI was misplaced as Everton quickly got going again, with goalkeeper Stefan Ortega quick to parry Jarrad Branthwaite's close-range header.
Tarkowski's departure left Everton's defence weaker and Keane's first offence resulted in a booking for a foul on Marmus – the first booking of the game, highlighting the team's overall lack of competitiveness.
Chelsea loanee striker Armando Broja's latest appearance, starting back-to-back games for the first time in three-and-a-half months but doing nothing to spur players towards a permanent move, saw him twice lose too easily to Josko Guardiola.
By then Beto had already lost the ball, but Abdoulaye Doucoure was the next threat, forcing Ortega to palm his corner kick towards the post.
And although Pickford came to Keane's aid at one point, he was unable to prevent either goal late in the match.
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