But City's night was tarnished after Ilkay Gundogan limped off on his first start since December following a serious knee injury.
City were dealt an severe blow when Gundogan went down after a Claudio Yacob challenge left the midfielder in heap on the deck.
Sane's ruthless finishes (3 and 77) either side of Claudio Yacob's close-range strike (72) were enough to send the visitors through despite a valiant display from West Brom.
The clash with last season's Premier League champions at Stamford Bridge could yet be deemed too soon for Gundogan but he remains confident the knock will not keep him out for long.
"I think it's a more motivation for everyone to show the coach that they want to play", Sane told City's official website. "Unfortunately the [foul] is tough. He will be back soon, hopefully", Guardiola explained, as quoted by ESPN.
And it could be even sooner than expected that we see the German worldwide back on the pitch. 'You can not imagine what it's like for months, every day alone or just with the physio not in the locker room.
Manchester City have confirmed that Ilkay Gundogan is expected to return to training in the next few days. "He was okay, he had a little bit of pain in his knee but tomorrow we are going to see and clarify what it is".
Pep Guardiola made no less than eight changes to his starting line-up that dismantled Watford at the weekend.
Leroy Sane had put them ahead in the third minute and scored the 77th minute victor to secure their place in the fourth round five minutes after Yacob had equalised.
Manchester City face Roy Hodgson's Crystal Palace on Saturday night.