Despite failing to win at home for the first time this season, Newcastle still extended their lead at the top of the Championship table to three points.
But this was by far United's sternest test of the season after trailing to an early Ben Watson goal.
Danny Guthrie, who had a nightmare match, saw a penalty saved before super-sub Harewood bundled in his first goal for the club on his home debut.
QPR went into the game having suffered only one defeat in their eight starts this season, away to Newcastle's opponents this Saturday, Bristol City.
Rangers started off well with some slick passing and control and it was little surprise when they went ahead after just seven minutes.
The pace of Wayne Routledge did the damage when his surging run found the on-loan Watson whose shot from outside the box deflected off Steven Taylor to beat Steve Harper.
United had a golden opportunity to draw level after 18 minutes when Taylor's volley from a free-kick was handled by Damion Stewart and referee Ron Shoebridge pointed straight to the spot.
With Shola Ameobi out injured, the spot-kick responsibility fell to Guthrie whose woeful effort was kept out by Radek Cerny diving to his right.
Guthrie could have redeemed himself on the half-hour mark when breaking in from the right but went for glory instead of choosing either Nile Ranger or Andy Carroll and Cerny easily saved.
Carroll then had an overhead kick cleared for a corner and, from the flag kick, headed beyond the keeper only to see Watson pop up on the line to head clear.
Carroll then whipped in a cross that posed problems and Ranger, at full-stretch, only managed to send his looping header inches wide.
A change was made at the break with Jonas Gutierrez replacing skipper Nicky Butt.
Rangers started the second half as they began the first, intent on going forward and almost doubled their tally when a Stewart header flew wide with Harper well beaten.
With the home fans becoming increasingly frustrated with their side, Rowan Vine could have compounded their problems with a great chance that he put wide.
Against the run of play it was all square after 69 minutes when Danny Simpson's inswinging cross from the right was headed across goal by Carroll and substitute Harewood - only on for eight minutes - bundled in his first goal for the club.
Harewood then put another opportunity wide, but a winner for Newcastle would have been an injustice.