The visitors controlled large parts of the first half and were worthy of their lead at the interval after Anthony Martial made it two goals in two games at the start of the new season.
However, Wolves responded impressively after the break and equalised through Neves, who added another long-range special to his growing collection.
Pogba had the perfect chance to maintain United's winning start to the campaign when he won a penalty, only to see his spot kick saved by Rui Patricio.
United handed Daniel James a full debut following his goalscoring cameo in last weekend's rout of Chelsea, and it was the visitors who made the more confident start to the match as they quickly took control of possession.
Chances were difficult to come by, though, and it was not until the 19th minute that either side had a clear-cut opening when Marcus Rashford turned Ryan Bennett inside out before delivery a cross which Martial was inches short of reaching.
The same two combined eight minutes later, and this time Martial made no mistake as he lashed a vicious finish into the roof of the net after being played through by his strike partner.
The France international's 50th Manchester United goal came with the very first shot of the match, and there were only two more before half time - neither of which seriously threatened either goal.
There was another nervy moment for the home side just before the interval, though, as an errant back-pass almost sent Martial through on goal, but Willy Boly recovered to make a crucial last-ditch challenge.
Nuno Espirito Santo introduced Adama Traore at half time in an attempt to spark his passive side into life, and the change almost paid immediate dividends when he began a move which ended with Raul Jimenez firing over when under pressure from Harry Maguire.
Jesse Lingard and Pogba then failed to threaten with shooting opportunities as the second half began to shape up into a much more entertaining affair, and Wolves came within inches of levelling nine minutes after the restart.
Wolves would not have to wait too much longer to restore parity, though, and the equaliser came in some style as Moutinho fizzed a ball out to Neves, who got it out of his feet before curling a pinpoint 25-yard strike in off the underside of the crossbar.
VAR momentarily threatened to rule the goal out for an offside in the build up, but the game quickly got back up to speed with Traore dragging another shot wide moments later.
United began to hit back shortly after the hour mark with Wolves clearing one dangerous cross into the box before Rashford fired a long-range free kick well off target, and the visitors were handed a golden opportunity to level things up 22 minutes from time.
Pogba won the penalty himself after going down under the challenge of Conor Coady in the area, but he subsequently saw his spot kick well saved by Patricio as he opted for power over placement.
The Frenchman then nodded wide moments later as he looked to atone for his miss, but Wolves were also looking for a late winner and Jonny Castro had an opening eight minutes from time which saw him plant a header straight at De Gea.
The hosts had another chance to win it at the death when they once again caught United on the break, and this time it was Leander Dendoncker who was the guilty party as he fired his effort over the bar.
The result means that United miss the chance to move top after two games, leaving Liverpool and Arsenal as the only teams with a 100% record at this early stage of the campaign, while the Red Devils are now five Premier League away games without a win.
Wolves, meanwhile, become the first team in Premier League history to come from behind and avoid defeat in three consecutive games against United as they made it back-to-back draws at the start of the new season.
WOLVES (3-5-2): Patricio; Bennett, Coady, Boly; Doherty (Traore 46'), Dendoncker, Neves, Moutinho, Jonny; Jimenez (Cutrone 91'), Jota (Neto 86')
MAN UTD (4-2-3-1): De Gea; Wan-Bissaka, Lindelof, Maguire, Shaw; Pogba, McTominay; James (Greenwood 89'), Lingard (Mata 81'), Rashford (Pereira 89'); Martialbody check tags ::