Coke struck in the 38th minute, producing an emphatic finish after bursting into the area following a lay-off from Marcus Tudgay.
Brighton had their chances to score with Elliot Bennett, Jamie Smith and Glenn Murray all going close.
Before the goal, Bennett tested Nicky Weaver in the Wednesday goal with a couple of efforts and Owls skipper Darren Purse saw his glancing header turned over by Casper Ankergren. Smith shot narrowly wide towards the end of the first-half and then struck the top of the bar with a volley soon after the re-start.
Coke threatened to find the net again when his shot on the turn was well saved by Ankergren.
As the visitors pressed for an equaliser, Murray saw his angled effort roll wide of the far post after being set-up by Smith. Marcus Tudgay and Neil Mellor both went close for Wednesday in the latter stages.