Moore directed a cross from Jay McEveley inside the left-hand post just when it seemed the Rams were heading for defeat.
The visitors will feel they had done enough to deserve victory and for most of the night they frustrated an out-of-sorts Derby team, who had trailed to a 13th minute goal from Martin Rowlands.
Much of Derby's play was disjointed and they failed to put QPR keeper Simon Royce under any sustained pressure.
With Danny Cullip marshalling the back four, Rangers pressed Derby all over the pitch and would probably have taken all three points but for a superb second-half save by Stephen Bywater.
The Derby keeper had been a spectator when Paul Furlong volleyed against a post and, as the defence stood still, Rowlands nipped in to poke the rebound into the net.
The closest Derby came before the break was a Steven Howard header which was cleared off the line by the outstanding Lee Cook and Rangers continued to impress in the second half.
Cook was causing the home side plenty of problems with his clever footwork and ability to run at players, but it was Steve Lomas who almost gave Rangers a two-goal cushion in the 58th minute.
QPR manager John Gregory held his head in his hands in disbelief, but his team still looked set for victory as Derby's attacks became increasingly desperate.
Royce had not had a save of any real note to make and time was running out for the home side when they finally broke through with two minutes remaining.
Howard did well to play the ball back out to McEveley on the left and he whipped the ball across for Moore to place a header just inside the left-hand post.
Derby now sensed that victory was on the cards, but although four minutes of added time was held up by the fourth official, they could not find another way through and Rangers deservedly left with a point to help their bid to stay in the Championship.