The German worldwide had looked sharp at The Hawthorns, setting up Leroy Sane's opener just three minutes into the game.
Sane, though, had other ideas and swept home the victor on 77 minutes to set up a fourth round contest with Championship side Wolves.
Guardiola will hope progress did not come at a cost after Ilkay Gundogan limped off 59 minutes into his latest comeback game.
Gundogan was forced out of the Carabao Cup win at the Hawthorns, his first start following a nine-month lay-off with a knee injury, prompting fears he had suffered another serious set-back.
City, however, broke to win the tie with 13 minutes left.
Gundogan played a role in Leroy Sane's opening goal, taking the initial shot which was blocked by West Brom keeper Ben Foster into the path of the Germany worldwide.
Them, from the resulting corner, Sane crossed for Eliaquim Mangala, whose header was beaten away by the diving Foster.
The German celebrates his victor with Gabriel Jesus.
His quick footwork outwitted Gareth McAuley on the edge of the West Brom six-yard box, but the winger could not keep down his shot and it cleared the crossbar. The ball had rebounded to him from a fine save by Albion goalkeeper Ben Foster who had beaten out a precisely guided effort from Ilkay Gundogan.
Claudio Yacob's tackle from behind in the 55th minute saw Gundogan signal to the bench to be substituted.
Guardiola added of Sane: "He played so well, not just in terms of goals - his second was fantastic - but how he kept the ball". So, thanks, "Guardiola said half jokingly".
Albion levelled on 72 minutes when Yacob scored his first goal in four years as he cashed in on slack marking to volley in Kieran Gibbs' corner.