The Hoops' freefall down the table continued in a poor affair at Bramall Lane that saw United come out on top from Norwood's eventual winner.
However, the pressure continues to rise on Mark Warburton, as his side sit bottom of the form table after just two wins from their last six games.
Story of the match
United began the game with a good intensity, as they looked to regain possession quickly from QPR and drive forward with it, forcing the visitors into committing needless fouls or giving away corners; this is how the Blades opened the scoring on nine minutes.
John Fleck drilled a corner into the feet of George Baldock, who let the ball roll through his legs and into the path of the on-rushing Norwood. The midfielder hit a low drive through a crowd of bodies that deceived former Sheffield Wednesday goalkeeper Keiren Westwood, who poorly threw a leg at the effort.
United continued to dominate the early play, but this time the Blades slowed down the movement of the ball and when they did lose it, the home side penned Rangers into their half and pushed them to cheaply concede possession.
On 17 minutes, United pushed again. A forward ball was collected by Oliver McBurnie, who held off Jimmy Dunne before attempting to thread in Morgan Gibbs-White, but Rob Dickie stepped out of the line of defence to thwart the chance.
Rangers were left scratching their heads and as they struggled to play through and around the Blades, so they attempted to go over; however, Lee Wallace’s superb execution was claimed by Wes Foderingham before Ilias Chair could capitalise.
On 25 minutes, the ever-electric Gibbs-White found the head of McBurnie with an outside-of-the-foot cross towards the back post, but the forward’s effort was a simple catch for Westwood.
Berge then had a sight on goal after a lung-busting run from midfield on the half-hour mark, but his drilled shot was sent wide via the slide of Yohan Barbet.
That half-chance excited the Bramall Lane crowd, but it soon turned into frustration as the tempo of the fixture fizzled out quickly, with not much of note happening until the final minute of the first half when Jeff Hendrick glanced a header wide of the target.
The first chance of the second period fell to the hosts, with John Egan steering wide from Norwood’s cross from deep.
Moments later, the Hoops threatened for a potential equaliser, but Chair dragged wide of Foderingham’s near post from the edge of the area.
On 52 minutes, Gibbs-White almost added a second with a sublime chip from 20 yards that had Westwood beat, but the effort fell just past the far post.
However, the fast start soon petered out as possession ebbed and flowed between the two sides, with little happening until the 74th minute when Lyndon Dykes’ delivery almost caught out Foderingham following a deflection from Egan. But the ball instead nestled on the roof of the net, much to the relief of the Blades' support.
Rangers pressed for a leveller, but it would be United who would have the next opportunity, with Gibbs-White scuffing a low shot into the hands of Westwood from the edge.
Gibbs-White then missed another great chance in the final minute of the game. The dynamic midfielder skipped past the challenge of Dickie before drilling wide of the far post.
However, the Blades held on to pick up a vital home victory.
Sheffield United: Wes Foderingham, George Baldock, Enda Stevens, John Fleck, Sander Berge (Ben Osborn, 83'), Oliver McBurnie (Daniel Jebbison, 73'), John Egan (C), Oliver Norwood, Ben Davies, Filip Uremović, Morgan Gibbs-White
Queens Park Rangers: Keiren Westwood, Lee Wallace (C) (Sam McCallum, 72'), Rob Dickie, Yohan Barbet, Lyndon Dykes, Ilias Chair, Sam Field, Andre Gray (George Thomas, 68'), Jimmy Dunne, Jeff Hendrick (Andre Dozzell, 68'), Albert Adomah
Unused Subs: Murphy Mahoney, Luke Amos, Charlie Austin, Dion Sanderson