Defensive errors cost Liverpool victory
The Merseysiders twice went ahead through Martin Skrtel and Luis Suarez on manager Brendan Rodgers' home debut, but failed to hold the lead on both occasions with costly errors at the back gifting Yaya Toure and Carlos Tevez goals.
It means Rodgers' wait for a first Liverpool victory will wait another week after his side lost 3-0 to West Bromwich Albion last weekend.
After a sluggish start, Liverpool began to dominate proceedings and should have taken the lead when Fabio Borini sent a cross from impressive 17-year-old Raheem Stirling just wide.
Carlos Tevez went close at the other end, his shot from an acute angle hitting the post and spinning back into play.
Manchester City's Vincent Kompany nearly steered the ball into his own net when an attempted clearance from a low Steven Gerrard drive went narrowly over the crossbar.
The hosts got the goal their pressure deserved from the resulting corner however, with Gerrard's cross being met with an unstoppable header from Skrtel on the edge of the six-metre box on 34 minutes.
City's equaliser was against the run of play when Toure took advantage of Pepe Reina and Martin Kelly's failure to clear a Tevez cross, sweeping the ball into the net on 63 minutes.
Liverpool took the lead back barely a minute later when below-par striker Suarez hit a stunning free-kick from 20 metres out, after Jack Rodwell was harshly judged to have handled a Gerrard shot.
City, who came from behind to beat Southampton 3-2 last weekend, again pulled level with Tevez's 100th goal in English football when Skrtel went from hero to villain, his poor back-pass being intercepted by the Argentine who calmly rounded Reina to roll the ball into the empty net on 80 minutes.
In Sunday's early match, Arsenal and Stoke City played out a 0-0 draw at the Britannia Stadium.
The result means Arsenal are still searching for their first goal of the season after they were held to a scoreless draw by Sunderland last weekend.