It was a bright start for the home side as Freddie Sears combined well with Murphy to force a corner, before teeing Darren Ambrose up for a shot that was turned away by Kieren Westwood inside the opening 10 minutes.
On the stroke of half time Murphy crashed a header against the post, but the Championship's top scorer didn't have to wait long for his goal.
It was the lively Sears who floated a back-post cross towards the waiting Murphy six minutes after the restart, and the striker made no mistake with the header, comfortably placing the ball beyond Westwood for his 19th goal of the season.
Half-time substitute Tyrone Mings then picked out Ambrose, but the midfielder could only send his header against the post.
Mings then curled an effort narrowly wide himself as the home side stepped up the pressure, and they were ahead just after the hour mark through their captain Luke Chambers.
It was a slick exchange of passes between Murphy and Kevin Bru that sent the French midfielder free, and his ball picked out Chambers who thumped a header into the back of the net.
Lee's speculative effort from range was as close as Wednesday came to rescuing a point as McCarthy's side comfortably held on to record their first win in four games.