Inigo Idiakez and Seth Johnson had over-run the Owls midfield at first, the Spanish midfielder pulling the strings as Wednesday crumbled before a second-half rally.
Derby had dominated the Hillsborough side for an hour. Dexter Blackstock, on loan from Southampton, should have added to his three-goal tally early in the game slotting the ball past the advancing Weaver from 12 yards only for it to cannon off the foot of the far post.
Weaver saved the Sheffield side after they were sliced open by a three-man move down the left flank. Full-back Peter Whittingham provided the final pass picking out Blackstock whose low drive was palmed away by the keeper.
Wednesday threatened in patches, Lee Peacock heading a Glenn Whelan cross over the top from close range before Idiakez caused more problems.
Weaver dealt with two corners whipped in by the Derby midfielder before Idiakez carved out another opening, his inch-perfect free-kick found Marcus Tudgay who pulled his header high over the target.
Derby, in total control, were caught out as the impressive Chris Eagles threaded the ball to Peacock but his low cross into the goalmouth had too much pace for 19-year-old Gabriel Agbonlahor as he raced into the box.
Tudgay broke the deadlock in the 41st minute with the Sheffield back four static as Whittingham crossed from the left, the unmarked Tudgay picking his spot, tucking his header into the top left-hand corner from five yards.
Midfielder Brunt fired his free-kick high and wide in the 49th minute and skipper Whelan had a shot beaten out as Wednesday mounted a second-half fightback.
Brunt grabbed the equaliser in the 69th minute, his fifth in seven games, chesting the ball in from close range after Graham headed on Peacock's cross before the unmarked Graham coolly slotted the winner past Lee Camp on 73 minutes.