Bury created a host of chances but their failure to capitalise on them proved costly when Coke burst into the left-hand side of the area and drilled a low shot beyond Cameron Belford.
Wednesday defender Mark Beevers went close to turning the ball into his own net soon after the goal and Bury's Andy Bishop saw his close-range effort come back off the bar.
Bishop twice went close in the first half, forcing Nicky Weaver to make a good save and then putting a shot on the turn wide.
Steven Schumacher, Ryan Lowe and Andy Howarth also missed the target as the visitors applied the pressure.
After Wednesday's Tommy Spurr struck the bar with a free-kick, Weaver came to the home side's rescue again, keeping out efforts from Tommy Lees, Danny Carson and Howarth.