Roberto Firmino looked to have sealed a potentially vital point for leaders Liverpool in the title race when he equalised just past the hour, but Sane struck soon after, confirming the Reds' first league defeat of the season in their 21st game and moving City to second and within four points of the summit.
Liverpool was mightily unlucky not to take an early lead when a defensive mix-up between John Stones and Ederson followed Sadio Mane hitting the post, but they survived and finished the first half well – Sergio Aguero netting in ruthless fashion.
Firmino restored parity in the 64th minute and City initially looked rattled, as if they could see the Premier League trophy moving out of sight, until Sane found the bottom-right corner to breathe life into their challenge once again.
A lively City start faded and they had a lucky escape in the 17th minute. Mohamed Salah released Mane into the box and the Senegal forward hit the post before Stones blasted his clearance against Ederson and hastily scrambled to clear off the line.
Vincent Kompany was fortunate to avoid being sent off for a wild lunge on Salah 14 minutes later and City capitalised on that let-off just before the break.
Bernardo Silva played a low cross from the left towards the near post, where Aguero nipped in front of Dejan Lovren and emphatically smashed past Alisson from a seemingly impossible angle.
Liverpool hit back 26 minutes from time, however, as Andrew Robertson latched on to a lofted pass and played a volleyed cross for Firmino to bury with a stooping header.
But City was not done.
It caught Liverpool on the break and Sane found the net via the right post after being played into the box by Raheem Sterling.
A Liverpool onslaught followed, with Ederson tipping a Salah effort wide, before Bernardo Silva and Sterling missed glorious chances to finish the Reds off – luckily for them, no fightback came and the champions held on.