The result leaves United supporters biting their nails, with the gap between the Sheffield side and third-placed Watford reduced to seven points following their home victory over Cardiff.
United fell behind with six minutes gone after a rare blunder from goalkeeper Paddy Kenny.
Furlong prodded a hopeful ball to Marc Nygaard and the Danish striker darted towards the area chased by defenders Chris Morgan and Chris Armstrong, but over-ran the ball.
The Blades shot-stopper dived but failed to snatch the loose ball cleanly as Nygaard finished off from six yards, for his ninth of the season.
Michael Tonge set-up United's equaliser in the 21st minute with a precise cross from the right, which £1.75million striker Ade Akinbiyi stretched to head beyond veteran keeper Paul Jones.
United swept ahead in the 29th minute as the London side failed to clear Chris Armstrong's in-swinging corner and the unfortunate Marc Bircham headed into his own net under pressure from home skipper Chris Morgan.
The Blades threw away a chance to kill off Rangers in the 49th minute after R's centre-back Ian Evatt hauled down Morgan as he tried to connect with an Armstrong free-kick.
Paul Ifill produced a tame spot-kick that was snatched on the line by the diving Jones.
And just seven minutes later Rangers made it 2-2, courtesy of another own goal. Matthew Rose stabbed a short corner to Lee Cook and his cross was met by Nygaard, but the ball hit the unfortunate Morgan and spun over the line.
Furlong added Rangers' third in the 74th minute, running on to a Cook pass and evading Morgan to coolly steer the ball wide of the advancing Kenny from eight yards.
The ex-Birmingham striker should have grabbed his second of the afternoon a minute later after Kenny failed to hold Cook's eye-catching strike, but from six yards he crashed the ball against the bar.
Armstrong had a chance to level matters seconds from time but pulled the ball wide from close range.