West Ham took the lead at the end of the first-half. Paul Pogba looked to have blocked Declan Rice's effort in the area with his face, but after review by VAR it showed that the United midfielder raised his arms and Michail Antonio made no mistake with his penalty.
The Reds responded after half-time and drew level when Anthony Martial linked up with Greenwood, who found the bottom corner of the net six minutes into the second-half.
United are up to third in the Premier League and know they need to avoid defeat against Leicester City on the final day of the season to secure a place among Europe's elite next season. West Ham are now unbeaten in their last three league games and secured their status in the top-flight after the point at Old Trafford.
Story of the game
After their loss against Chelsea in the semi-final of the FA Cup at the weekend, the Reds made four changes to the team with Pogba, Greenwood, Anthony Martial and Timothy Fosu-Mensah coming into the starting line-up.
United settled quickly and started the game on the front foot from the beginning. With just two minutes on the clock, Bruno Fernandes played a perfectly weighted pass to Martial, who fired a shot across goal and drew a low save from Lukasz Fabianski.
The West Ham goalkeeper was called into action again, just under a minute later. Fernandes linked up with Greenwood this time and the young striker was denied by Fabianski.
Ole Gunnar Solskjaer’s men were given a warning sign when Aaron Cresswell’s whipped cross flew straight across the six-yard box and evaded several bodies before going behind.
The Reds continued to threaten going forward with Martial the latest United player to go close to opening the scoring. Pogba and Nemanja Matic linked up to find the Frenchman, who was unable to hit the target from the edge of the area.
West Ham started to grow into the game and open up from the half-hour mark and their first opening of the game came in the 31st minute.
Mark Noble played a short corner into the path of Jarrod Bowen from the right and the cross into the box found the head of Angelo Ogbonna and the defender headed over a good chance.
Rashford tested Fabianski with a rasping drive from the left before Fosu-Mensah guided Brandon Williams’ cross wide at the back post. However, it was the visitors that closed out the opening 45 minutes on the front foot.
De Gea was called into action for the first time when Ben Johnson’s cross was headed by Victor Lindelof onto Antonio and the United goalkeeper had to react quickly to make the save.
West Ham had appeals for handball turned away as Rice’s shot struck Pogba in the box. VAR looked at the decision and the ball had, in fact, hit the arms of the French midfielder and the visitors were awarded a penalty.
Antonio slotted the spot-kick into the bottom right corner of the net, sending De Gea the wrong way and put the Hammers ahead at half-time.
Solskjaer got the reaction he was looking for and the Reds started the second-half with more urgency and they equalised six minutes after the restart. Martial found Greenwood with a neat give-and-go and the latter broke into the penalty area before firing low into the bottom corner.
West Ham immediately went on the attack as Bowen got the better of Williams down the right side before seeing his shot deflect to the edge of the area to Rice, whose volley didn’t trouble the United ‘keeper.
Pogba went close for United on the hour mark with a speculative effort from range, which went just over the bar, but the visitors continued to threaten down at the other end.
Bowen gained a yard on Williams in the area and drew a terrific save from De Gea, who was equal to the shot and tipped the ball over the bar in the 62nd minute.
The Hammers certainly weren’t settling for a point and continued to threaten in the final third. Rice fired an effort inches wide of the post before Noble’s free-kick was glanced over the bar inside the six-yard box by Antonio.
Odion Ighalo replaced Rashford for the final stages of the game and almost made an instant impact in the 85th minute. Greenwood skipped past a challenge and broke into the box before pulling the ball back, but the substitute was unable to wrap his foot around the ball and fired a yard wide of the post.
Solskjaer's side stepped up their intensity in the closing stages, but were unable to create another opening and know a point against Leicester on Sunday will secure a place among Europe's elite next season.
What's next for both teams?
The Reds have dropped points at home for the second consecutive game, but despite the result against West Ham, their fate is now in their own hands going into the final day of the season. United looked like they ran out of gas midway through that second-half, but finished on the front foot.
The Hammers are now safe for another season, two wins and a draw from their last three games has secured West Ham's status in the Premier League. David Moyes' side will be at home to Aston Villa on the final day of the season.