Simon Grayson's side spent much of the afternoon on the back foot but returned home celebrating their first victory at Bramall Lane since 1978 following the striker's clinical 69th-minute finish.
The result, which was greeted with a chorus of boos by the home crowd, leaves United languishing in the League One relegation zone and Weir, a former Scotland international, potentially battling to save his job.
United, handing full debuts to new signings Marlon King and Ryan Hall, created the better chances of the first-half.
Jose Baxter forced Declan Rudd to make a flying save at the near post after unleashing an angled drive while the Preston goalkeeper was called into action again soon when Tony McMahon went close from a set-piece.
Febian Brandy and Matt Hill both tried their luck from long-range as United continued to press but the latter was responsible for the slip which allowed the visitors to take the lead against the run of play.
Kevin Davies, profiting from the defender's stumble, found Beavon lurking inside the area and he cut inside to beat George Long from close-range.
It was tough luck on United, who had earlier seen a strong appeal for a penalty turned down when Brandy's shot appeared to be handled by Bailey Wright.