Rangers looked to be in cruise control after Marc Nygaard took advantage of a sleepwalking United defence just six minutes in, but Ade Akinbiyi levelled out of nothing with his first Blades goal at Bramall Lane.
Just before the half-hour, Marc Bircham - who scored the decisive goal when QPR beat United in August - crumbled under the pressure of Chris Armstrong's corner to head a calamitous own goal.
Paul Ifill missed a penalty that would surely have put the game out of reach and Rangers rallied as Nygaard's header forced Chris Morgan to turn the ball into his own net and Furlong finished off a swift counter to turn the game on its head.
Blades boss Warnock responded to the wasteful finishing that almost cost his side their Sheffield derby win last weekend by handing a debut to loan signing Geoff Horsfield.
Bircham, meanwhile, was making his return from injury for Rangers, while Furlong's consolation strike at Crystal Palace was rewarded with a return to the starting XI.
The visitors had proved something of a bogey side for United at Bramall Lane, with last season's 3-2 success one of only two home wins the Blades had managed against QPR in 10 attempts.
And, after a tame opening, another upset looked on the cards when Nygaard picked up the ball on the halfway line before bearing down virtually unchallenged on goal.
Paddy Kenny advanced to smother the ball at the Dane's feet but it rebounded back off the striker who tapped home his sixth of the season.
Rangers were looking extremely comfortable as the Horsfield-Akinbiyi partnership failed to sparkle, although the on-loan West Brom man won a free-kick from which Armstrong tested veteran goalkeeper Paul Jones.
And having been no threat whatsoever, Akinbiyi suddenly snatched a fabulous equaliser, virtually back-heading Michael Tonge's inswinging left-wing cross beyond the despairing Jones.
The goal brought United to life, Ifill causing havoc down the right touchline and forcing some desperate Rangers defending, and the pressure eventually told as Armstrong's inswinging corner was headed into his own net by Bircham.
Neill Collins almost marked his home debut with the third when he glanced Tonge's free-kick narrowly wide, while Jagielka sliced his finish after the visitors twice failed to clear Ifill's low cross.
Despite being under the cosh, Lee Cook almost brought Rangers level with a terrific free-kick Kenny did well to get hands on and the winger was at it again moments later with a brilliant run before feeding Furlong, whose finish was too close to the Blades keeper.
But their inability to defend crosses cost them again just after the break when Jones flapped at Armstrong's free-kick and in the confusion Ian Evatt felled United skipper Morgan - making his 100th league start for the club.
The referee immediately pointed to the spot but Ifill's penalty lacked power and Jones got down well to parry to his right.
It proved a costly miss as Rangers played a short corner to Cook, whose inswinging cross was headed on target by Nygaard and - despite being on the line - Morgan could only turn the ball into the net.
Warnock threw on Danny Webber, the pacey striker back after six weeks on the treatment table, while Rangers were forced to replace Nygaard with Sammy Youssof after the big striker went down injured.
But it mattered not as a United attack broke down and Rangers countered, the ball eventually fed to Furlong, who turned Morgan inside out before finishing left-footed past Kenny.
The veteran striker missed a glorious chance to seal the points when he rattled the crossbar after Kenny spilled Cook's drive and Youssof should also have finished it but Kenny saved well one on one.
Armstrong should have levelled in stoppage-time but Rangers held on to become the first side to beat United home and away this season.