His 25-yard shot ensured a 1-0 victory which was made even tougher for the home side who were forced to play the final 40 minutes with 10 men after the dismissal of midfielder Andrew Crofts.
Brighton boss Sami Hyypia gave debuts to keeper David Stockdale, veteran defender Aaron Hughes, and strikers Chris O'Grady and Shamir Fenelon.
O'Grady, captured from Barnsley for £500,000, quickly made his presence felt and put a left-foot shot on the turn wide in the 12th minute.
The Owls defence was put under pressure when O'Grady later burst through and debutant keeper Keiran Westwood came to the visitors' rescue by diving to smoother a goal-bound left-footed strike from winger Kazenga LuaLua.
LuaLua, hoping to put a history of injury problems behind him, tried his luck from long range before O'Grady was crowded out following a through ball from Jake Forster-Caskey.
Fenelon, who made his League debut while on loan at Torquay last season, forced Westwood to save low down when he got a shot away on the half hour.
O'Grady headed over from a corner by Forster-Caskey before midfielder Coke struck to give Wednesday the lead against the run of play four minutes before the interval.
Brighton paid the penalty for not closing down Coke tightly enough as he rifled home a left-foot shot into the top corner from 25 yards.
LuaLua dragged a shot wide from the edge of the area, as the Seagulls failed to respond before the break
Striker Chris Maguire failed to make the most of a sharp chance for Wednesday in front of their 2,500 travelling fans when he fired over after an accurate cross by Joe Mattock.
The Seagulls were reduced to 10 men with the sending off of midfielder Andrew Crofts on 50 minutes.
The Welshman received a straight red card from referee James Linington for a high tackle on defender Sam Hutchinson.
Brighton had a let off when Jacques Maghoma got clean away down the flank but crossed wastefully with the Seagulls defence at full stretch.
The Seagulls should have pulled level when skipper Gordon Greer set up O'Grady just inside the area, the former Barnsley marksman went for placement rather than power and keeper Westwood pushed his shot round the post.
Wednesday threatened to extend their advantage when Kieran Lee burst into the area and his flighted ball just eluded the diving Coke.