Pep Guardiola’s side were in danger of losing more ground in the title race following a second-half collapse at the Etihad Stadium.
Foden and Josko Gvardiol put City two up before the interval, but third-bottom Leeds hit back through Dominic Calvert-Lewin and Lukas Nmecha.
Foden clinched a nerve-jangling victory that lifted City into second place, four points behind leaders Arsenal, who travel to third-placed Chelsea on Sunday.
It was a much-needed win after consecutive defeats against Newcastle last weekend and Bayer Leverkusen on Tuesday.
“It was a relief,” Guardiola said. “The game was not perfect in the first half but it should be over with the chances we had.
“Football is emotions. We conceded the second and after we made a step-up. In the end it’s the quality of Phil.”
City had made their worst start to a campaign for five years and their second-worst under Guardiola.
But they took just 59 seconds to move in front as Foden fired a close-range strike in off the bar.
Leeds had no answer to City’s slick start and Gvardiol looked to have put the hosts in complete control with a tap-in from Nico O’Reilly’s header in the 25th minute.
Calvert-Lewin made an immediate impact after coming off the bench, reducing the deficit with a clinical finish after pouncing on Matheus Nunes’s woeful pass inside the City area in the 49th minute.
Leeds equalised in the 68th minute when Nmecha’s penalty — awarded for Gvardiol’s foul on Calvert-Lewin — was saved by Gianluigi Donnarumma before the forward recovered to score from the rebound.
In stoppage-time, Foden cleverly made room in the area for an ice-cool finish as Guardiola breathed a sigh of relief.
At the Tottenham Hotspur Stadium, Frank’s men made a disastrous start as Kenny Tete put Fulham ahead before Harry Wilson’s superb strike punished a calamitous blunder from goalkeeper Guglielmo Vicario.
Tottenham were booed off at half-time and although Mohammed Kudus pulled one back after the interval, there were more jeers at the final whistle.
With only three wins in their last 13 games in all competitions, Tottenham have slumped to 10th place.
Tottenham are without a home league victory since the opening weekend of the season, wining only four of their 10 matches in front of their fans in all competitions this season.
Frank was critical of the fans who taunted Vicario, saying: “I didn’t like that our fans booed at him. They can’t be true Tottenham fans because everyone supports each other when you’re on the pitch and we do everything we can to perform.
“I’m fine with them booing after the match, no problem, but not during. That is unacceptable in my opinion.”
Newcastle celebrated boss Eddie Howe’s 48th birthday with a 4-1 rout of Everton at the Hill Dickinson Stadium.
A first away league win since April 7 was the ideal present for Howe, whose side made an explosive start when Malick Thiaw headed home from Lewis Miley’s corner after just 52 seconds.
It was the fastest goal in the Premier League this season.
Beaten in their previous four away games, Newcastle netted again in the 25th minute when Everton goalkeeper Jordan Pickford allowed Miley’s shot to squirm past his woeful attempted save.
Nick Woltemade made it three with a cheeky chip over Pickford in the 45th minute.
Thiaw’s 58th-minute header added to Everton’s misery and Kiernan Dewsbury-Hall’s 70th-minute eye-catching strike was little consolation for David Moyes’s men.
Brentford striker Igor Thiago reached 11 league goals this season as his double fuelled a 3-1 win against second-bottom Burnley in west London.
Sunderland climbed to fourth place after fighting back from two goals down to beat Bournemouth 3-2 at the Stadium of Light. (AFP)
