Ole Gunnar Solskjaer's side took the lead through Anthony Martial's second Premier League goal of the season in the first half.
Ruben Neves hit a brilliant equaliser for Wolves after the interval before France star Pogba was denied by Patricio.
It was a curious decision for Pogba to take the spot-kick as Marcus Rashford had scored a penalty in last weekend's 4-0 victory over Chelsea and also memorably netted the Champions League last-16 winner from the spot against Paris Saint Germain last term.
Pogba has now missed four Premier League penalties since the start of last season.
United would have gone top of the table with a win, but instead they were left to reflect on an inconsistent performance that underlined why Solskjaer still has a lot of work to do to turn his team into title contenders.
While there were moments of promise in attack and defence, United didn't take their chances and crumbled too easily when they came under pressure in their second game of the season.
At an average of 24 years and 173 days, Solskjaer had picked United's youngest Premier League starting line-up since the final day of the 2016-17 campaign.
Daniel James made his maiden start for United after the close-season signing from Swansea scored his first goal as a substitute against Chelsea.
There was no place in the matchday squad for Chile forward Alexis Sanchez, who has been linked with Inter Milan.
Solskjaer has described James as United's 'x-factor', but the 21-year-old didn't win any friends when he was booked for diving in a failed attempt to win a free-kick from referee Jon Moss.
United threatened for the first time when another of their young guns Rashford spun past Ryan Bennett on the left side of the Wolves area and crossed towards Martial, who did not make enough contact as the ball flashed past him.
It was a warning Wolves failed to heed as United took the lead with an incisive raid in the 27th minute.
Breaking inside from left, Luke Shaw found Jesse Lingard who moved the ball onto Rashford.
Rashford slipped a perfectly-weighted pass to Martial in the Wolves area and the French forward blasted a superb strike into the roof of the net without breaking stride.
Solskjaer had called for Martial to be more clinical and, his message received loud and clear, the Norwegian clapped approvingly on the touchline.
Martial's goal took him to 50 for United in all competitions since his move from Monaco in 2015.
Martial should have doubled United's lead on the stroke of half-time when he pounced on Bennett's misplaced pass and eased past Willy Boly, only to tread on the ball when he seemed certain to score.
Well marshalled by Harry Maguire, United denied Wolves a single shot on target in the first half.
But Wolves were revitalised after the interval by the introduction of the speedy Adama Traore.
Suddenly, Maguire and company did not look so solid and Raul Jimenez glanced a free-kick against the far post with United keeper David De Gea rooted.
Wolves' pressure paid off in the 55th minute when a short corner was worked to Neves on the edge of the area and the Portugal midfielder curled a sublime strike into the top corner.
Neves's eye-catching effort survived a VAR review for offside in the build-up and he celebrated his 10th goal from outside the area since joining Wolves.
United had a golden chance to reclaim the lead when Martial found Pogba in the Wolves area and the France midfielder was clipped by Conor Coady.
Pogba and Rashford appeared to discuss who should take the spot-kick.
Whoever made the final call, it proved a bad move as Pogba's powerful drive was superbly pushed away by Patricio.