The Spain forward held his nerve in the fourth minute of added time to convert his first goal of the season after Leeds substitute Joe Gelhardt had gone down under Wolves defender Nelson Semedo’s challenge.
Marcelo Bielsa’s depleted Leeds had laid seige on Wolves’ goal for much of the match after Hwang Hee-chan had given the visitors a 10th-minute lead.
Wolves appeared to have held on for their fourth straight win with a battling backs-to-the-wall display in another highly-charged encounter at Elland Road but Rodrigo’s last-gasp spot-kick denied them.
Leeds barely deserved to trail 1-0 at half-time after a much-improved display in which their renowned energy levels were restored.
Rodrigo headed Raphinha’s corner wide after Bielsa’s side had burst out of the blocks, clearly intent on putting last week’s no-show in a 1-0 defeat at Southampton behind them.
But Wolves delivered a thumping body blow out of nowhere in the 10th minute to snatch the lead from their first attack.
Semedo mugged Jack Harrison on the right edge of the area and Raul Jimenez’s scuffed shot hit Hwang and fell kindly for the South Korean forward to turn the ball home from close range.
Prompted by the rangy brilliance of Brazil winger Raphinha, Leeds made most of the running but save for Mateusz Klich’s curled effort, easily snaffled by Wolves goalkeeper Jose Sa, they had little to show for their efforts.
Leeds coped more comfortably without midfield linchpin Kalvin Phillips – the England international returned to the bench after a calf strain – but Patrick Bamford’s continued absence leaves a gaping hole up front.
Harrison blazed high and wide and Raphinha curled his shot off target after cutting inside.
Leeds were dealt another hefty blow early in the second half when Raphinha was caught late by Wolves defender Romain Saiss and hobbled off. He was replaced by Crysencio Summerville.
Bielsa sent on young striker Joe Gelhardt for Klich in the 63rd minute as Leeds regained some momentum.
Saiss just got his head to Daniel James’ cross as it appeared destined for Summerville at the far post and Roberts’ glancing header drifted wide.
Pascal Struijk headed Stuart Dallas’ free-kick straight at Sa and the home faithful cranked up the volume in their bid to lift their side.
Leeds appealed in vain for a penalty when Rodrigo went down under Max Kilman’s challenge and Gelhardt’s thumping drive was tipped over by Sa.
Gelhardt snatched at another chance, blasting over, but in the dying moments the former Wigan starlet tumbled in the area under Semedo’s challenge and Rodrigo coolly converted after referee Robert Jones had pointed to the spot.