The midfielder hit the target from 25 yards after Steve Kabba had cancelled out Alan Quinn's 47th minute opener with the Hillsborough side chasing their first league win at Bramall Lane for 36 years.
And with 12 minutes to go substitute Wayne Allison snatched the third to keep Neil Warnock's side on course for the play-offs and Wednesday still locked in a relegation battle.
Wednesday keeper Kevin Pressman who has made a habit over the years of frustrating United, had brought off a couple of vital stops in the opening minutes.
Dean Windass, signed from Middlesbrough 48 hours earlier on a free, cut in from the left flank and fired in a fierce shot which crashed against the woodwork. The ball fell to Peter Ndlovu eight yards out and his header was palmed over the top by Pressman.
The former Bradford striker went close again on 32 minutes after Wednesday striker Michele Di Piedi was caught in possession. Windass, picked out by Brown, who produced a shot on the turn from a tight angle, the ball flying inched over the target.
Wednesday broke the deadlock in the 47th as Quinn ran on to a harmless looking ball headed clear by the Wednesday defence. As a space opened in front of him the Wednesday midfielder raced into the box unchallenged before tucking the ball past the advancing Paddy Kenny from 16 yards.
United were back on level terms in the 62nd minute, evergreen Stuart McCall stretching to reach the ball and pick out Kabba who twisted to stab a low shot beyond Pressman from ten yards.
Shefki Kuqi missed another opportunity heading over from four yards before Brown snatched the lead with a sensational goal in the 65th minute, the former Manchester City man running on to a half clearance to slam the ball into the far corner with an angled drive from 25 yards.
Allison grabbed a third 12 minutes from time, Kabba running along the right flank before crossing for the big striker to slot home from close range.
Said a delighted United boss Warnock: "I think this is the best night of my life. It feels better than beating Liverpool and Wednesday last year.
"We rolled our sleeves up and showed tremendous character. We played some great football, normally you don't get that in derby games.
Owls boss Chris Turner shouldn't be disappointed. His team battled. If the club can survive this year he will bring in some new players and it will be a different Wednesday you will see next season.
Added Turner, a former Wednesday keeper: "When you get beaten by Sheffield United you are going to be disappointed. The gulf between the two teams was there for all to see and we were under pressure for 75 minutes.
"Neil has had three and a half years to build a team that has got a great chance of promotion.
"At the moment United have the upper hand, but we will be back. I feel certain we can get out of the bottom three and we will be back at the top in a few years."