Wanderers were on the brink of joining Norwich at the summit after Nick Blackman’s own goal gave them the lead in a see-saw match with just six minutes remaining.
But Murray had other ideas and with a minute to go crashed in his third goal since his deadline-day switch from Crystal Palace on loan.
Earlier in the second half on Sunday, a crazy three-minute spell saw Wolves come from behind to lead 2-1, only to immediately concede an equaliser to Jake Taylor.
The result still saw Wanderers, who are challenging for back-to-back promotions following last season’s League One title success, climb back up to third – but they have won only once on the road so far this term.
George Saville blew a great chance to settle those away-day nerves in the fifth minute when he was played clean through only to shoot straight at Royals keeper Adam Federici before Lee Evans skied the rebound.
But Reading, themselves aiming for a return to the Premier League after suffering relegation last year, made the breakthrough after 19 minutes when Wolves were caught napping by a short-corner routine, leaving Oliver Norwood free to pick out defender Michael Hector to power in a near-post header for his first Royals goal.
Murray came close to doubling the advantage with a long-range shot which flew just wide and Norwood fired narrowly over from 30 yards just after the interval.
But Wolves grabbed a surprise equaliser in the 51st minute when Bakary Sako’s low cross eluded three Reading defenders and left former Royals trainee James Henry with a simple finish at the far post.
Three minutes later the visitors had turned the game on its head, when Obita’s miscued clearance fell to Evans who swept the ball home from 10 yards.
But Reading amazingly hit back within 60 seconds, Cox charging at the Wolves defence before finding Taylor whose angled shot flew through Carl Ikeme’s legs and into the net.
Yet Wolves regained the advantage in the 84th minute when Sako’s corner skimmed off the head of Dave Edwards and was nodded into his own net by Blackman despite Norwood’s attempted clearance on the line.
However, Wolves failed to heed the lessons from earlier in the half and were pegged back again when Jamie Mackie teed to Murray to crash in the equaliser and earn a deserved point for the hosts.