Preston boss Simon Grayson paid tribute to his players after their fourth consecutive victory and first at Bramall Lane since 1978.
Beavon pounced during the second-half of a match which leaves Sheffield United still languishing in the relegation zone and David Weir, who only took charge in June, battling to save his job.
But Grayson has no such concerns having seen complete a historic smash-and-grab raid.
"I'm delighted," he said. "This is a terrific arena for football and the opposition have got a wealth of experience and some very talented players.
"But we showed a really strong spirit out there. It was an excellent all-round professional display from us. You saw some of our flair players sacrifice themselves to help out at the back. And that tells you all you need to know.
"We withstood everything they threw at us and limited them to very few clear cut chances."
"Stuart took his goal ever so well," Grayson added. "He showed brilliant composure.
"He didn't have too many openings but you now that, when he gets one, he is going to put it away and that's exactly what he did.
"I made it clear beforehand that records are there to be broken. As we've just proved.
"Overall I'm not going to get carried away by anything. We are going well and confidence in the dressing room is very good as you'd expect.
"But nothing, nothing at all, is won at thuis stage so we've just got to keep it going."