Hart was the shock choice to replace Jim Magilton, who left the west London club last week following his alleged dressing room bust-up with Akos Buzsaky.
QPR co-owner Flavio Briatore is desperate for instant success and following his nightmare spell as Portsmouth manager, Hart was handed the poison challis of becoming the eighth manager since the Italian multi-millionaire took charge of QPR back in September 2007.
But on the evidence of this display, Hart will have to use all his bargaining skills in the January transfer window if he is to steer QPR into play-off contention.
QPR fans are fed up with the managerial merry go round at Loftus Road, but gave a warm welcome to Hart at the start of the game and must have thought he was finally the man to turn their season around after they opened the scoring after just two minutes.
United failed to clear a corner and Mikele Leigertwood fired a low drive which took a wicked deflection and flew past United keeper Mark Bunn.
It was the first goal United had conceded in over four-and-a-half hours and was an extra special moment for Leigertwood, who was scoring against his former club.
Hart was not celebrating for long as United quickly fought their way back into the game and grabbed an equaliser on eight minutes.
Ben Watson played a horrendous back pass straight into the path of Toni Kallio, who crossed the ball into Richard Creswell and he poked a close-range shot into the corner of the net.
Hart was looking for an immediate reaction and fair credit to the QPR players for mounting continued pressure on the United goal.
Adel Taarabt, on loan from Tottenham, almost put QPR back into the lead on 20 minutes when he made a clever turn inside the box and fired the ball just past the post.
Sheffield United players were doing their best to contain Taarabt but he continued to cause all kinds of problems for the United defence and on 44 minutes came close again when he struck a 25-yard shot just over the bar.
Despite the home side's dominance, it was United who could have grabbed a second on the stroke of half-time when Stephen Quinn's deflected cross fell perfectly for Jamie Ward inside the box but he struck his volley inches past the post.
As temperatures dropped below freezing, Hart would have instructed his players for more of the same in the second half.
His players almost responded with the second goal on 59 minutes. Patrick Agyemang's cross fell perfectly to Jay Simpson whose volley fell just wide of the post.
QPR had strong claims for a penalty turned down on 65 minutes after Chris Morgan appeared to bring down Jay Simpson inside the box, but referee Neil Swarbrick ignored the strong pleas from the home supporters and waved for play to continue.
The home side could have snatched a dramatic victory in injury time when Wayne Routledge cross fell to Watson whose header struck the top of the bar.
United held on and will be happy to come away with a point which extends their unbeaten run to seven league games and keeps them on the fringes of the play-offs.