In a game billed as 'must-win' for the champions, Sergio Aguero powered them into a half-time advantage before Roberto Firmino hauled the Reds on terms midway through the second half.
Sane was the match-winner 18 minutes from the end, sliding a slow shot across goal and in off the base of the upright.
It condemned Liverpool to a first league defeat of the season - just as they had done to City almost a year ago - and means the Merseysiders' advantage is cut to just four points from City, who leapfrogged Tottenham into second spot.
The game was high intensity and high on quality from the word go, with both teams moving the ball at pace and with purpose.
It was Liverpool that came closest to a breakthrough on 18 minutes.
The front three of Firmino, Mohamed Salah and Sadio Mane combined, with the last-named beating Ederson inside the box only to see his shot cannon back off the post.
With John Stones attempting to clear, the City goalkeeper dived and inadvertently sent the ball towards his own net, leaving the England defender to somehow clear the deck with Salah lurking and ready to convert.
That let off stifled City briefly but they got their noses ahead five minutes before the break.
After Aguero's penalty claims were denied, Bernardo Silva recycled and his low cross found the Argentine.
Having nicked in ahead of Dejan Lovren, the City striker lashed a venomous shot into the roof of the net from an improbable angle.
Liverpool responded after the break and got their reward on 64 minutes.
Trent Alexander-Arnold's cross from the right was cleverly diverted back across goal by opposite full-back Andy Robertson, giving Firmino the easiest of tasks.
Pep Guardiola's men summoned another rally and eight minutes later they got the all-important third goal.
A swift counter attack found Raheem Sterling and, with Aguero's decoy run taking away defenders, he found the composed Sane, the German blasting a low shot in off the base of post. Salah had Liverpool's best chance to level, his left-footed shot palmed around the post by Alisson before Sterling almost got the goal he craved late on, his shot from 18-yards flying agonisingly wide.
City held on for three vital points, igniting the title as Liverpool suffered their first body blow of the season.