In a nervy opening, anxious Wednesday were gifted an early opportunity when Leon Knight was chopped down by Robbie Pethick 25 yards out. However, Knight took the free-kick himself and hit a sloppy shot straight at the keeper.
A second free-kick attempt by Knight was poor and the Brighton keeper made a comfortable save.
Brighton, with only one away win to their credit, caught Wednesday napping after 19 minutes.
Steve Haslam's tame headed clearance dropped at the feet of Steven Sidwell who whipped in a testing cross and Hart was left unmarked to head down past Kevin Pressman.
It could have been worse for Wednesday six minutes later when Bobby Zamora produced a clever turn on the left before cutting in and finding Sidwell but he hoisted his first-time shot over the top from close in after a timely tackle from Tony Crane.
Wednesday carved out two promising openings but skipper Trond Egil Soltvedt smashed a 20-yard shot over the bar after a spell of pressure.
Two minutes later Craig Armstrong's shot on the turn was parried by a diving save from Michels Kuipers before Knight lifted the follow-up over the top.
Wednesday's best opportunity to level came after a double substitution by manager Chris Turner on the hour.
Strikers Lloyd Owusu and Knight were replaced by Finnish forward Kuqi and Dutchman Gerald Sibon.
The pair had an immediate impact. Kuqi hustled his way into the area and caused panic but it was Sibon who forced a save from Kuipers with a shot. He then saw another snapshot blocked on the line before Kuqi turned the follow-up wide.
Brighton had two late opportunities to put the game beyond doubt.
Zamora again did well bursting into the area and when his cutback was cleared as far as Hart he threaded in a neat pass for Graham Barrett.
However the substitute scuffed his shot and in the dying moments Zamora blasted wide.