The Hillsborough outfit, who last reached the fourth round in 2001, were more than a match for Roy Hodgson's side before they tired late in the game.
Injury ravaged Wednesday, down to the bare bones and with five Academy players on the bench, should have snatched the lead in the opening minutes.
Teenage striker Luke Boden, playing in an unaccustomed midfield role, drilled in a low ball from the left flank.
It fell nicely for Johnson who scuffed his shot as Mark Schwarzer pulled off a one-handed save.
The Championship side paid the price on 12 minutes as Fulham snatched the lead in clinical fashion.
Danny Murphy picked out Andrew Johnson with the sweetest of passes as the striker coolly slipped the ball under the advancing Lee Grant.
Wednesday were back on level term on 21 minutes with a stunning strike.
James O'Connor squared the ball to full-back Tommy Spurr, who struck from 30 yards with the ball flying into the top left-hand corner.
Grant, who has caught the eye of several Premier League clubs, foiled Andrew Johnson by tipping his close-range header over the bar before Wednesday squandered another chance.
Wade Small picked out Boden on the left and the 19-year-old produced an inch-perfect cross which Jermaine Johnson headed over from six yards.
Jermaine Johnson was the culprit again on 42 minutes. This time Francis Jeffers unselfishly played the ball to the striker, who fired over from 15 yards.
Grant pulled off a brave save saving at the feet of Clint Dempsey to block his close range strike, before the Fulham midfielder produced a dipping shot from 20 yards which was snapped up by the Wednesday keeper.
Substitute Erik Nevland fired wildly over the top before Grant foiled Dempsey again as the London side finished the stronger.
Nevland shot inches wide before Andrew Johnson wrapped up the game by steering his cross past Grant from ten yards with two minutes to go.