United have moved into fifth place in the top-flight standings, but Wolves drop down to ninth position despite extending their unbeaten run in the league to four matches.
However, the visitors were made to rue those missed chances moments later as Fred benefitted from a neat layoff from Paul Pogba to drill a low shot into the bottom corner of the net, registering his first goal since arriving from Shakhtar Donetsk.
Wolves continued to have their moments but the remainder of the first half belonged to United, despite their struggle to create clear-cut chances in the final third.
Alexis Sanchez sent a free kick high over the crossbar but in the final seconds before the break, it took a stunning Rui Patricio save to prevent Fred's set piece from 25 yards sneaking inside the near post.
Both teams returned for the second half with the intention of going onto the front foot, and a block from Patricio prevented Sanchez from doubling United's advantage from close range.
However, Wolves were soon level. A ball from the byline found Jimenez in the middle and after the Mexican teed up Moutinho who was waiting on the edge of the area, the midfielder did the rest with a sublime curling effort into the top corner from 18 yards out.
The visitors were in the ascendancy with both Costa and Diogo Jota trying their luck from distance, but neither Pogba or Jesse Lingard were shy from hitting long-range efforts at the other end with Patricio tipping the latter's strike over the crossbar.
United began to take control as the match entered the final 20 minutes, and Marouane Fellaini should have done better with a header when being found at the back post by Luke Shaw as Patricio made a timely block.
The home side dominated the closing stages, but in the first minute of added-on time substitute Adama Traore saw a well-struck shot kept out by De Gea at his near post.
Traore had two more chances before the full-time whistle as United left space at the back, but despite failing to net with two speculative attempts Wolves will be delighted having recorded another impressive point against one of the division's biggest clubs.
MANCHESTER UNITED (4-2-3-1): De Gea; Valencia, Lindelof, Smalling, Shaw; Fred (Martial 63'), Fellaini; Lingard (Pereira 75'), Pogba, Sanchez (Mata 63'); Lukaku
WOLVERHAMPTON WANDERERS (3-4-2-1): Patricio; Bennett, Coady, Boly; Doherty, Neves, Moutinho (Saiss 80'), Jonny; Jota (Gibbs-White 87'), Costa (Traore 75'); Jimenez