A win for Wolves would have seen them join current league leaders Norwich City on 20 points, but Kenny Jackett's men fell behind to a well-worked goal by the hosts after 19 minutes.
Jordan Obita and Oliver Norwood exchanged passes from a short corner, before Michael Hector beat his marker to power home a header from the latter's cross at the near post - the defender's first goal for the club.
Murray sent an effort fizzing wide of the left-hand upright from 25 yards soon after as the hosts looked to double their adavantage.
However, Wolves scored the next goal six minutes into the second half, against the run of play - Bakary Sako's low cross worming its way through the Reading defence for James Henry to tap in at the far post against his former club.
And the match was turned on its head three minutes later when Obita inadvertently played the ball to Lee Evans in the area, with the Wolves man taking full advantage.
Jake Taylor's instant reply after an excellent throughball from Simon Cox looked like earning the hosts a point, but Wolves were not done.
A Sako corner somehow found its way over the line six minutes from time. The ball was initially thought to have come off Wolves midfielder Dave Edwards, but replays suggested it was actually Nick Blackman who got the final touch.
The ball also appeared to have been hacked off the line, but the goal was given, with a second view indicating the ball clearly crossed the line.
Wolves looked destined to keep pace with Norwich at the top of the table, but Murray's late deflected strike from 20 yards out earned Reading a deserved share of the spoils.
