Chelsea won 1-0 at 10-man Swansea City to stay within two points of leaders Liverpool, who they visit in two weeks time.
A Chelsea victory in that game could let in City, who trail Liverpool by seven points but have two games in hand, one against bottom club Sunderland on Wednesday.
Liverpool have not finished top since 1990, two years before the Premier League began, but Raheem Sterling and Martin Skrtel put them 2-0 up inside 26 minutes before David Silva led a City revival in the second half.
The Spaniard reduced the deficit and an own goal by Glen Johnson levelled matters before Brazilian midfielder Philippe Coutinho drove in the winner after a mistake by City captain Vincent Kompany 12 minutes from time to secure Liverpool's 10th successive victory.
Liverpool midfielder Jordan Henderson was sent off in stoppage-time and will miss three of the four remaining games, including the potentially decisive visit of Chelsea.
Manager Brendan Rodgers, who has revitalised the team in two seasons since taking over, hailed "a remarkable performance".
"I thought we were incredible today. We showed so much spirit and quality. We have been put under pressure in a very big game but you can see from the quality of our football that we are coping with it," he said.
It was an emotional day for the club, who commemorated 25 years since the Hillsborough disaster in which 96 of their supporters died at an FA Cup semi-final. There was a minute's silence before kickoff, after which the home fans roared their team into a sixth-minute lead.
Luis Suarez outmuscled his marker Gael Clichy and fed Sterling, the teenager cleverly dummying Kompany and England goalkeeper Joe Hart before scoring.
Twenty minutes later Hart saved well from a header by unmarked Liverpool captain Steven Gerrard but from the resulting corner Slovak defender Skrtel headed in.


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