Bright joined Sheffield Wednesday in September 1992 in a cash plus player exchange deal involving fellow striker Paul Williams that was rated at a total transfer value of £1,375,000. The Wednesday players were experienced and at the peak of their careers; manager Trevor Francis was eager to win trophies at Hillsborough and felt Bright would prove a good partner to club stalwart David Hirst, particularly with the highly talented Chris Waddle in midfield. After scoring six goals in the League Cup, Bright played in the League Cup final, which ended in a 2–1 defeat to Arsenal. Wednesday also reached the final of the FA Cup, where they would again face Arsenal, with Bright scoring an extra-time header against derby rivals Sheffield United to secure victory in the semi-finals. The final proved to be a disappointment however, as he controversially elbowed Arsenal defender Andy Linighan in the face, causing a broken nose. Later, with the scores level in the last minute of extra time, a heavily bandaged Linighan out-jumped Bright from a corner to score the winning goal. After the game, Bright phoned Linighan to apologise.