But the Blades responded smartly and equalised in first half stoppage time when Cresswell stepped across his marker to send his looping header into the top left-hand corner.
In an end-to-end second half the best chance came 10 minutes from full-time when United's Ched Evans thumped an incredible 35-yard free-kick onto the underside of the bar but neither side could grab the winner.
On a bitterly cold afternoon at Bramall Lane, the home side dominated early on and twice came close to breaking the deadlock.
Jamie Ward produced a superb right-wing free-kick that fell straight onto the head of Chris Morgan, but the Blades skipper headed wide from six yards in the sixth minute.
Three minutes later Henri Camara showed superb skill to slip into the box before pulling back the ball from the byline to the unmarked Stephen Quinn, who rattled the crossbar when it looked easier to score.
United continued to dictate the tempo and were looking likely to make their pressure count through Camara, who was running QPR ragged.
But as the half reached the midway point Rangers began to get a foothold in the game.
Rangers' attacking duo of Patrick Agyemang and Simpson were looking dangerous, with the latter forcing Mark Bunn into a sprawling save down to his right from 30 yards in the 27th minute.
Minutes later Routledge's goal bound shot was blocked but the deflection fell kindly to Alejandro Faurlin, who forced Bunn to palm wide with a 25-yard swerving drive.
The visitors were beginning to control proceedings and made the pressure count in the 38th minute.
Routledge's right-wing cross skidded across the six-yard line to Simpson, who guided the ball across keeper Bunn into the bottom right-hand corner to break the deadlock.
United immediately tried to equalise and got the response they desired on the stroke of half-time.
Cresswell shook off his marker in the box to head over the outstretched arms of keeper Radek Cerny and level the tie heading into the break.
United took the game to their Championship rivals in the second half.
Camara saw his veering shot palmed away by Cerny before Jamie Ward's fizzing effort curled just wide of the top angle.
But as the half progressed, the match descended into a scrappy, end-to-end affair with both sides having chances in equal measure.
Damion Stewart came close for QPR when he bulldozed his way through the box to head just wide for Rangers, which was shortly followed by Camara forcing Cerny to save well from close range at the other end.
Other than Evans' rasping free-kick in the 80th minute the game petered out, resulting in a replay at Loftus Road in mid-January.