For the second week running since winning promotion, City were disappointing, and Peter Taylor took action by making two half time substitutions, introducing Marc Joseph and Andy Hessenthaler.
Hessenthaler should have scored after coming on, but from the resulting corner Wednesday gave away a penalty because of a clear handball. Stuart Elliott blasted it low down the middle for his 29th goal of the season.
Ben Burgess came off the bench in the 72nd minute to a standing ovation, replacing Nick Barmby. He didn't see much of the ball, but did drag a speculative shot wide.
Boaz Myhill pulled off a great fingertip save to stop Adams's superb 20-yard volley, but then made a howling error to allow Wednesday to score an injury time winner. Alex Bruce put in a long throw and Myhill was unable to capture it with an attempted punch, and it was knocked in on the line.
The attendance of 24,277 was the biggest ever for a City game at the KC Stadium, and the players were presented with their League One runners-up medals after the match. Meanwhile, Sheffield Wednesday go 4th and secured a play-off place.
City line-up (4-4-2): Myhill; Stockdale (Joseph 45), Cort, Delaney, Edge; Ellison, Ashbee, Lewis (Hessenthaler 45), Elliott; Barmby (Burgess 72), Fagan.
Un-used subs: Duke, Dawson.