City had dominated the opening period with a sparkling display of attacking football, but paid for missed opportunities as the Baggies fought their way back into the tie after the break - before Sane struck a superb second of the evening late on.
The Blues go into the last 16 of the competition as a result, there will be concern for Ilkay Gundogan who limped off after a heavy challenge with an hour played.
City may have made eight changes from the side that beat Watford so convincingly at the weekend, but the starting XI at The Hawthorns looked just as slick as the one that finished at Vicarage Road.
The Blues knocked the ball around for fully three minutes, making 54 touches before West Brom’s Ben Foster parried a shot from Ilkay Gundogan and Sane lashed home the rebound from close range.
It was an incredible start to the game yet fully deserved for a City side oozing confidence.
With skipper Yaya Toure’s calming influence in midfield, Gundogan looking sharp and inventive and the energy and pace of Bernardo Silva, Leroy Sane and Raheem Sterling, the Blues had sprinted out of the blocks in impressive fashion.
And the lead was nearly doubled twice in the space of a minute midway through the first half.
First Gundogan found himself free on the edge of the box and the Germany international was only denied by a last-gasp challenge that resulted in a corner that Eliaquim Mangala saw a header well saved by Foster.
It was one-way traffic and Gabriel Jesus’ trickery then saw him wriggle free of his marker and fire a shot over – he will feel he should have scored - before Sane was denied by the onrushing – and very busy – Foster.
City were swarming around the Baggies’ box and the hosts looked a little bewildered by the incessant pressure they were being put under and had this been a boxing contest, the referee would have probably stopped the bout.
But West Brom’s threat from set-pieces meant that so long as they were just a goal behind, they were still in the game.
In first-half added time, City again came close as Gundogan found Sterling in acres of space and the England winger drove into the box before curling a shot towards the top corner before being clawed away by the impressive Foster.
The hosts weren’t quite as under the cosh as they had been in the opening minutes of the second half, but a clumsy challenge on Gundogan ended the midfielder’s evening and hopefully his early exit was no more than a precaution given his recent return from long-term injury.
West Brom came within a whisker of levelling on 69 minutes when James Morrison almost squeezed a low shot inches wide of the post and moments later Sane should have doubled City’s lead but chose to cross instead of shoot when through on goal.
It proved a costly miss as the Baggies levelled – from a set-piece – when Yacob headed home a Kieran Gibbs corner.
The Blues had paid the price of not killing the game sooner and within minutes should have been behind when Matt Phillips’ low cross found Hal Robson-Kanu but the Wales striker somehow dragged his shot wide from eight yards out.
It was quite a turnaround, but City responded in the best manner possible with Bernardo Silva driving into the Baggies’ half before finding Sane who collected the pass before cutting inside and curling a beauty past Foster to restore the Blues’ lead.
There were one or two nervy moments - including another amazing miss by Robson-Kanu - in the time that remained, but the victory was deserved by the Blues, roared on by 4,000 travelling fans.
The misses by Robson-Kanu were crucial. The first at 1-1 with West Brom in the ascendency would have surely swing the game Albion’s way - then another shot wide when clear on goal in the dying seconds - were let-offs for the otherwise impressive Blues.
City Matchday App star man: Leroy Sane
What it means…
City go into the last 16 of the competition and will take on Wolves at the Etihad - date and time TBC...
City return to Premier League action on Saturday with a home game against bottom side Crystal Palace.
There’s no let up, either, with the second Champions League group stage match against Shakhtar Donetsk visiting the Etihad the following Tuesday.
To mark his Man of the Match display, we are giving fans a chance to win a Sane shirt.