Dominic Calvert-Lewin bagged a brace and Daniel Cockerline also got on the scoresheet as United scored three times in the opening 23 minutes to take a firm grip of the game. In the second-half, the visitors overcame a couple of individual errors to secure a first win of 2015.
United started brightly and deservedly took the lead in the 8th minute. After a corner wasn't cleared properly, Hinchliffe whipped in an inviting cross that was met powerful by the head of CALVERT-LEWIN who gave the goalkeeper no chance from close range.
The Blades doubled their lead on the quarter hour, Hinchliffe was again instrumental in the build-up and after a neat flick by Brooks, CALVERT-LEWIN bent his shot past the outstretched arm of the home goalkeeper to notch his second of the game.
A very good start for the visitors became an excellent one moments later as Hinchliffe chased what looked like a lost cause before robbing the defender of possession and teeing up COCKERLINE to make it 3-0 with just 23 minutes on the watch.
That was arguably the last meaningful action of the first-half, with the Owls looking shell-shocked.
The second-half was a scrappy affair. Just after the hour mark only the width of the post denied Brooks entering a contender for goal of the season with a dipping volley from 25 yards out.
But United then handed the hosts a lifeline. A aimless ball ran through to Ramsdale, but instead of picking the ball up the goalkeeper failed to control with his foot and that allowed the chasing Owl to slot home into an empty net, this sparked a siege on the Blades goal.
In their next attack, Cockerline lost his man from a free-kick and from being seemingly in control of the game, it was now 3-2. Wednesday continued to press and thought they had levelled in the 90th minute, only to be denied by the offside flag.
United finished the game with Calvert-Lewin being deployed at centre-back after an unfortunate injury to Adebowale, who took a nasty whack when putting his head in where it hurts.
Coach Travis Binnion was understandably delighted with the win, he remarked: "The scoreline at half-time reflected our dominance and the game only became a contest after we made a couple of poor decisions in the second-half.
"The likes of Calvert-Lewin, Evans and Cockerline made a difference and gave us that bit more experience that we have been missing and it
emphasised the importance of scoring first, especially when you are on top, we haven't done that in recent weeks.
"Hopefully this will spark a successful and strong end to the season and of course it is always nice to win a Steel City derby."
Blades: Ramsdale, Brown, Kelly, Ishmail, Evans, Adebowale (Slew 76), Brooks (Ramsey 81), Gordon, Calvert-Lewin, Cockerline, Hinchliffe.
Unused: Hutchinson, Charlesworth, Perryman.