A goal from Aguero and assist helped the Premier League champions cancel two first half goals from the Championship side.
Grimes stepped up and calmly dispatched the penalty, sending the keeper to his left and placing the ball into the top-left corner.
He takes it first time, opening his body up and curling a right-footed effort around Ederson and into the back of the net.
69 minutes on, Aguero took a shot and it was blocked but comes back out to him, and the Argentinian finds Bernardo Silva, who arrowed a sublime strike into the far corner with his left foot to pull one back for City.
8 minutes later, center referee awards City penalty after tussle between Carter-Vickers anfand Sterling for the ball, the England winger goes down in the box.
2 minutes to stoppage time, Aguero grabs the winner for City. Bernardo Silva curled in a superb cross from the left and there was a suspicion of offside, but the linesman didn’t flag and the striker produced a superb diving header past a despairing Nordfeldt at his near post.
Next on the block for Swansea will be Nottingham Forest in Championship, while City will face Fulham at Craven Cottage after the international break.