Asaba, replacing the suspended Ade Akinbiyi, made his first League start since December 6 and took his chance in a thrilling West Midlands derby.
Walsall played a full part in the entertainment and would have salvaged a point if the offside Jorge Leitao had not got the final touch to Gary Wales' late header.
A mixture of slack marking, poor goalkeeping and moments of quality saw the sides share four goals in an eventful first half.
Darel Russell gave City and eighth minute lead when his 25-yard drive slipped through the clutches of Andy Patterson before entering the net.
Paul Merson showed his class five minutes later as he delivered a perfect centre for Leitao to bury a firm header beyond Neil Cutler.
But the nervous Patterson could only parry a shot from Kris Commons in the 36th minute and Asaba rammed the rebound into the roof of the net from six yards.
Walsall were giving as good as they got and their second equaliser was rolled in at the near post by impressive debutant Chris Taylor after Gary Birch had swept past John Halls.
Patterson gave his dented confidence a much-needed boost at the start of the second half when he made a splendid save from Lewis Neal. And the goalmouth action continued thick and fast as Merson drove a 25-yard shot against the post before Stoke regained the lead for the third and vital time in the 56th minute.
Gifton Noel -Williams held off three Walsall challenges before turning in an intended shot which Asaba headed home with Patterson wrong footed.
There were further opportunities at both ends before Walsall staged a late rally which should have brought a third equaliser two minutes from time.
A well-placed header from substitute Wales was about to cross the line when Leitao, ruled offside by a linesman, fired in.