Dawson dived to the left to refuse David McGoldrick's 15-minute kick.
Owl boss Jos Luhukay was under pressure after a shocking series of four consecutive defeats, but his men were more brave than the Blades.
Luhukay had called on fans to support Dawson's controversial pre-match election – and his faith was rewarded.
The Dutchman has selected Dawson ahead of Keiren Westwood, whose name was recently sung by Owls fans in support.
Dawson is in shape after rescuing a penalty that had been lost to Norwich 4-0 in the previous game.
This match was similar to the last Steel City derby here, with Sheffield United dominating in the 0-0 but not breaking through.
The men of Chris Wilder missed the chance to go to the top despite their positive attitude as Michael Hector and Tom Lees shut them out.
Wednesday had the worst defense in the division this season with 30 goals, which did not show up here.
The visitors have just proved the old Derby saying that the form came out of the window as they bravely beat and hit gaps.
Desperate Luhukay grabbed his defense again with three center-backs and wing-backs to stop the damage.
The derby atmosphere was usually electric and United looked pumped out, which quickly led to an early opening.
John Fleck threw a goal to the gate, but Dawson threw the ball impressive forward.
The Wednesday goalkeeper was about to accomplish even greater heroics when he saved United's penalty.
The spot kick was awarded when Mark Duffy drove forward but was knocked down by Morgan Fox.
McGoldrick scored his penalty on the left side of the keeper on Wednesday, who jumped and coolly pushed the ball away.
The Wednesday, however, remained the bottom line and soon needed central defender Hector to deny McGoldrick again.
The game was played at a fast pace and the Blades had another near loss before the break when Billy Sharp shot forward.
United continued to dominate at the beginning of the second half, but the pace of the game had dropped.
The best hope for the owls looked like Adam Reach was producing a characteristic moment of magic, but he scanned his first chance.
Reach showed his better qualities as he blocked an effort from Fleck when United's frustration continued.
Ex-owl Leon Clarke was struck by United in the last round, but even he could not help himself as Wednesday survived the late pressure.