City then dominated proceedings but Aguero was thwarted by Nordfeldt before wild efforts by Iheanacho and Jesus Navas.
Their wastefulness, which drew forlorn stares from Aguero, was punished in first-half stoppage time when Ayew restored parity in the most fortuitous of circumstances.
City defender Nicolas Otamendi was asking for trouble when he felled Swansea skipper Leon Britton with an ill-judged lunging challenge that earned a caution.
There was further punishment moments later as Ayew set his sights on goal with a 25-yard free-kick and found Lady Luck was smiling on him as Fernando’s head diverted the set-piece past the wrong-footed Hart.
Despite the set-back, City were back on the front foot immediately after the break and Eliaquim Mangala almost put the visitors back in control when he rose to meet Kevin De Bruyne’s cross but his header grazed the crossbar.
Iheanacho and Aguero spurned further opportunities as City went for the kill but Federico Fernandez served notice of their opponents’ ongoing threat when he headed Stephen Kingsley’s corner just wide.
That was as close as Swansea got to ensuring Pellegrini’s tenure at City ended on an agonising low, but their lacklustre display underlined the size of the task facing Guardiola when he heads to Manchester from Bayern Munich.