Billy Sharp, who scored 30 goals last season, continued that fine form to head the only goal of the game.
Leon Clarke produced a fine cross to captain Sharp who made no mistake in powering it past Brentford keeper Daniel Bentley.
Clarke then battled his way into the box and produced a similar cross straight to Sharp, but this time Bentley saved comfortably.
United had two goals disallowed for off-side from Clarke and Sharp while Bees striker Vibe also had one ruled-out.
Lasse Vibe and debutant Neal Maupay had chances to grab a late point for the visitors but were denied by Kieron Freeman and Jamal Blackman in goal.
Wilder said: "First and foremost I must talk about our support for the last six years. We've had it tough in this city, a football city as it is.
"That is a reward for the outstanding support, loyalty and love of their football club that they've shown through thick and thin.
"I've got to say, I thought that one [the atmosphere] was right up there today. Outstanding effort against a brilliant Brentford side.
"The roar that greeted us, the positivity allowed the players to lift their game at times when they were very, very tired. Sheff United fans, they deserved that result today.
"We've all got a point to prove, haven't we? Whether we've not managed in the division or been out of it, players, coaches and staff, are we good enough?
"Billy's goalscoring record speaks for itself and I think we've created chances today. We didn't play particularly well with the ball today - we can play a lot better."