Anthony Martial's 50th United goal opened the scoring in the 27th minute, the France forward reaching the milestone by converting the first shot of the game.
Wolves levelled in the 55th minute with a goal that survived lengthy VAR checks for offside, Ruben Neves smashing home a 20-yard drive after a clever set-piece routine.
Pogba had the best chance to take all three points but saw his spot-kick kept out by Patricio, with United unable to make amends for two defeats at Molineux last term.
Daniel James was booked for his second dive in the 24th minute as both sides, preferring to play on the break, felt their way into the match.
United broke the deadlock soon after, patient possession on the left leading to Marcus Rashford sliding a pass through for his strike partner, with Martial lashing home a first-time finish.
Wolves hit back after the break though, the half-time introduction of Adama Traore giving United a fresh problem to handle, with a glancing Raul Jimenez header hitting the inside of the post.
The equaliser came moments later. A short corner was well worked by Joao Moutinho to Neves, lurking with intent on the edge of the penalty area, and he crashed a trademark strike in off the underside of the crossbar.
RÚBEN NEVES TAKE A BOW! pic.twitter.com/RUDjFiUdXq— NBC Sports Soccer (@NBCSportsSoccer) August 19, 2019
United should have retaken the lead in the 68th minute but Pogba, having won a penalty for Conor Coady's clumsy challenge, failed to beat Patricio from the penalty spot.
RUI PATRÍCIO SAYS NO ❌ pic.twitter.com/sUnNxHFAeq— NBC Sports Soccer (@NBCSportsSoccer) August 19, 2019
A stretched finale saw chances at both ends, with Luke Shaw denied at the death as neither side could find a late winner.
What does it mean? Wolves still a menace for top-six rivals
Europa League duties could hinder Wolves' attempts to break into the top six this term but Nuno Espirito Santo's side look set to bloody some more noses in 2019-20 having matched United.
The Red Devils, who would have gone top of the league with a win, have now failed to win any of their past five away games in the Premier League.
Neves unstoppable from outside the box
United knew what Neves was going to do when the corner routine worked the ball to his feet 20 yards out. Defenders scrambled to close the former Porto captain down but he still found enough time and space to whip home his 10th goal from outside the area since joining Wolves in 2017.
James has to cut simulation out
There is no doubt the raw pace of Wales winger James makes him one of the more thrilling additions to the Premier League this season but he must learn to stay on his feet. Reputations are built quickly at this level and they can be hard to shake.
Ole Gunnar Solskjaer's side host Crystal Palace in the Premier League on Saturday. Wolves go to Torino for the first leg of their Europa League play-off on Thursday, then a busy start to the season continues when they host Burnley in the Premier League three days later.