City finally faltered at a ground where the team has a long history of struggles -- with just one league win in 37 years -- and was ultimately undone by nine minutes of mayhem, when Liverpool hounded City's defense and scored three goals.
With the score at 1-1, Roberto Firmino deftly chipped goalkeeper Ederson Moraes to regain the lead for Liverpool in the 59th minute. Sadio Mane smashed a rising shot high into the net in the 61st, and Mohamed Salah capitalized on a weak clearance from Ederson to score from 45 meters (yards) in the 68th.
City threatened an amazing comeback after goals by substitute Bernardo Silva, in the 84th, and Ilkay Gundogan, in the first minute of stoppage time, but Liverpool held on in a nervy finale.
"This was a historical game you will talk about in 20 years because it looks like City will not lose another one this year," Liverpool manager Juergen Klopp said.
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"People watched this game all over the world and this is why -- take your heart, throw it on the pitch and play like this."
A match away to Liverpool always looked like being the toughest of City's remaining 16 games and so it proved. City's lead stayed at 15 points, and it could be down to 12 if United beats Stoke today.
"People asked if everything was over and done before Christmas, and now it's January and it's not," Guardiola said. "We have a lot of games to win to be champions."
A triumph in 2003 was City's only win at Anfield since 1981, and Guardiola's team played at times as if it knew the recent history of the fixture.
Brazilian's hard work off the ball that created Liverpool's goal.
Firmino dispossessed Fernandinho, allowing Alex Oxlade- Chamberlain to take up the loose ball, run forward and send in an angled drive that flew low past goalkeeper Ederson.
Sterling was jeered relentlessly and even forced off the pitch at one point by marker Andrew Robertson. Later, he squared up to Robertson after a late clip of his heels and was then booked for a late challenge on Georginio Wijnaldum.
Liverpool didn't deserve to be out of the lead and stormed back in a stunning nine-minute spell around the hour mark when City's defense fell apart.
Firmino barged aside John Stones and chipped Ederson to make it 2-1 in the 59th. Moments later, Mane curled a shot against the post but was more precise the following minute when he lashed home a left-footed piledriver from the edge of the area after Nicolas Otamendi was dispossessed.
"We lost a little bit of our control," Guardiola said. "We were involved in the environment of Anfield for many, many reasons."
Guardiola tried to encourage his players from the sidelines as Liverpool's whole team celebrated inside a raucous Anfield, and City almost recovered.
Silva buried a low shot after a mazy run from Gundogan, who then poked in from eight meters for 4-3. There was to be no equaliser this time, though.
"At the end of the season," Sane said, "when we win the Premier League, nobody cares if we lost a game.
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