Man City restore six-point lead, United held by Watford.
Manchester City re-opened a six-point lead in the Premier League but needed a late goal at Everton and a slice of luck to keep their destiny in their own hands.
Phil Foden struck the only goal in the 82nd minute to break Everton's resistance but City breathed a sigh of relief when Rodri got away with a handball soon after with VAR failing to award Everton a penalty.
Manchester United's top-four hopes were dented as they were held to a 0-0 draw by relegation-threatened Watford.
Denmark international Christian Eriksen made an emotional return to competitive action as a substitute for Brentford, eight months after suffering cardiac arrest at Euro 2020.
But Brentford went down 2-0 at home to Newcastle United as their transformation continued under Eddie Howe.
Leeds United's slide continued as they suffered a 4-0 home loss by a Harry Kane inspired Tottenham Hotspur.
With Liverpool playing in Sunday's League Cup final against Chelsea, Man City knew only victory against Everton would keep the title race in their own hands.
With Everton repelling them it seemed that Pep Guardiola's side were about to open the door wide open for Liverpool, but Foden pounced late on to turn in Bernardo Silva's cross.
City moved to 66 points from 27 games with Liverpool on 60 having played 26 games. The two sides meet at the Etihad Stadium in April.
"We were searching. We tried and we were there. We earned the good fortune. Everton has top players. Their quality is there, we knew it," Guardiola said.
Everton are now only one place and one point above third-from bottom Burnley who completed a good week with a 1-1 draw with Crystal Palace at Selhurst Park.
Burnley have 21 points with Everton on 22, both teams having played 24 games. Leeds have 23 points in 16th place and have played two games more while Brentford have 24 but have played three games more than Burnley and Everton.
Watford, in 19th place, have 19 points after battling for a point at Old Trafford.
Cristiano Ronaldo hit the post in the fifth minute and had an effort disallowed for offside but United struggled to create enough clear cut chances against Roy Hodgson's side.
Bruno Fernandes was unable to convert two good opportunities in the first half and Anthony Elanga was off target with the best opportunity after the break.
United remain fourth but are only two points above Arsenal who have three games in hand.
"We did everything apart from score. It's hard to take that result," United's interim manager Ralf Rangnick.
"We need to be sharper in front of goal, you can hardly create more chances than we did today."
Eriksen's return to a football pitch was a happy moment in a bleak weekend for news but he could not inspire a Brentford comeback after coming on as a substitute.
"If you take away the result. I'm one happy man. To go through what I've been through, being back is a wonderful feeling," said the Dane.
Brentford had Josh Dasilva sent off after 11 minutes for a bad tackle and Newcastle took advantage with Joelinton and Joe Willock scoring before halftime as Newcastle climbed up to 14th place, having been bottom only three weeks ago.
Leeds fell apart at Elland Road as Matt Doherty and Dejan Kulusevski put Spurs in the driving seat inside 15 minutes.
Kane volleyed home a third in the 27th minute, making Leeds only the second side in Premier League history to concede three or more goals in five consecutive matches in the competition after Sunderland in November 2005.
Son Heung-min completed the rout late on.
Burnley earned a point at Palace as home defender Luka Milivojevic turned a drive from Aaron Lennon into his own net.
Aston Villa won 2-0 at Brighton & Hove Albion thanks to goals from Matty Cash and Ollie Watkins.
Many players showed their support for Ukraine as the Russian invasion continues. At Goodison Park Everton's team walked out draped in Ukraine flags while Man City's players wore T shirts emblazoned with "No War" slogans.
- Reuters