The seconds were ticking away in added time when the former Watford striker got the decisive touch in the penalty area after a Keith Gillespie free-kick.
Neil Shipperley had done the same trick for United in the first half, his acrobatic strike in the closing seconds cancelling out the visitors' lead.
Preston had rocked the Blades with the opener in the 35th minute when Paul McKenna's free-kick from 30 yards out evaded Paddy Kenny in the home goal.
Blades boss Neil Warnock made two changes after his side's 2-1 defeat at QPR, drafting in defender Craig Short for his home debut and bringing Michael Tonge back into midfield, with tonsilitis victim Lilian Nalis out of action.
United were quick to show and Carlo Nash had to punch clear from a Paul Ifill cross in the first minute, and they threatened when Webber won a corner after his left-footed shot was blocked by defender Claude Davis.
Jon Harley swung in the kick, but Shipperley was adjudged to have fouled the goalkeeper and the danger was cleared.
Shipperley, already on the scoresheet this season after a goal against Burnley, then fired in a right-footed shot which cleared Nash's crossbar as the Blades pressed for an opener, with Tonge drilling a left-footed effort wide.
Leigh Bromby was another to miss the target, this time from close range, and there was little to be seen of Preston as an attacking force until they broke the deadlock.
Shipperley, pressed into more action than boss Warnock wanted at this stage of the season, was taken off after 62 minutes, with Paul Shaw taking his place.
Preston, who had made four changes, saw more of the action in the second half and in a spell of pressure both David Nugent and Danny Dichio got in shots, with Kenny saving from the latter before he was replaced by Patrick Agyemang.
Warnock threw on winger Gillespie to replace Ifill with 20 minutes to go, before Harley's left-foot volley brought a save from Nash.
Phil Jagielka was the next to test the Preston rearguard but his shot following a free-kick went over the top as the Blades went for a winner with the minutes ticking down.
The midfielder had a chance to win the game with a good chance from close in after Gillespie's cross, but he put the ball wide.
Then, just as it looked as if Preston would hold out, Webber stepped in with a dramatic intervention to seal the points for United.