The hosts took the lead after just 30 seconds when Yannick Bolasie got to the byline before crossing to Murray, who poked home.
As the first half drew to a close it was the visitors who threatened most, with on-loan Barcelona frontman Rodri twice going close.
Despite the fact that the Owls couldn't draw level before the interval, they didn't take long to equalise after the break, with Antonio getting on the end of a Miguel Llera delivery.
Palace retook the lead seven minutes before the whistle when Wilfried Zaha did well on the right before picking out Murray in the middle, who tucked home his second.
The win sees Palace pick up their first points of the season and move off the bottom of the table, while Wednesday drop to sixth.