Inspired by Gundogan’s goals and Phil Foden’s stylish display, City moved 10 points clear of fourth placed Liverpool after their first win at Anfield since 2003.
The troubled champions were blown away by Pep Guardiola’s side, who became the first team to score four times in a league game at Anfield in 12 years.
City hold a five-point lead over second placed Manchester United and have a game in hand on their local rivals as they eye a third title in four seasons.
They have won 14 successive games in all competitions to equal the record for an English top-flight club, set by Preston in 1892 and Arsenal in 1987.
“We have done brilliantly so I cannot deny how happy we are. To come here and finally win, it’s very important,” Guardiola said.
“I’m so grateful to my players but this is now the past. In February no one is champions.”
Liverpool are in disarray after losing three consecutive home league games for the first time since 1963.
They have won just two of their last nine league matches as injuries and fatigue take their toll.
“We made two massive mistakes, that is clear. There is not a real explanation. Maybe he had cold feet? It sounds funny but it could be,” said coach Jurgen Klopp of the two poor clearances from Alisson that led to City goals.
“They used them and 3-1 down in a game like this, when you are that long level, it is tough to take.”
Gundogan blasted a 38th minute penalty into the Kop after Fabinho’s trip on Raheem Sterling.
But the Germany midfielder got back in Guardiola’s good books in the 49th minute with a close-range finish after Foden’s shot was parried into his path.
Liverpool were gifted a penalty in the 63rd minute after City defender Ruben Dias allowed Mohamed Salah to latch onto Trent Alexander-Arnold’s long pass and responded by pulling him down.
Salah took the penalty himself, showing Gundogan how to do it as he shot high into the net for his 22nd goal this season.
Alisson made an even bigger mistake than Dias in the 73rd minute when the Liverpool keeper’s sloppy pass was intercepted by Foden.
Foden made a superb run through the heart of the Liverpool defence before cutting back for Gundogan to finish from virtually on the line.
Alisson’s nightmare wasn’t over and in the 76th minute he again conceded possession with a woeful pass.
This time, Bernardo Silva was the beneficiary and he chipped a cross to Sterling, who headed into the empty net.
England midfielder Foden capped the rout in the 83rd minute, cut in from the right to blast past the shell-shocked Alisson.