United had the majority of possession and twice as many shots as their London opponents but could not find a way past goalkeeper Adrian.
It was a game befitting a contest nearing the end of the season though Luke Shaw did hit the post in the first stanza for United.
West Ham remain 15th, but just two points behind 10th place.
On Sunday, Manchester United play the final match of their season at Old Trafford against Watford, while West Ham host Everton at home.
That victory also helped secure European football, but the move to the London Stadium, sold to supporters as West Ham's chance to compete with the Premier League elite, has so far proved a false dawn.
Manchester United manager Jose Mourinho has, on several previous indications, made comments that give a strong indication of a coming summer clear out.
After cleaning out half the Avengers, Thanos arrives to take over Fortnite
We're glad they did though, the cross-over is a super fun addition to the game. To be clear, this is not a simple skin players are trying to take control of.
Jesse Lingard was lively, picking up good spaces in front of the Hammers' low block and forcing Adrian into two good saves from distance, but there was no cohesion with his strike partner Alexis Sanchez.
He nearly gifted a goal to West Ham in the first half after he sliced a clearance back into the six-yard box, and one fan said that he's a "disaster waiting to happen".
In the absence of the injured Lukaku, Lingard appeared to be United's most likely scorer, though he could not beat Adrian with a 20-yard strike in the 55th minute.
A lacklustre United display was reflected in Pogba again failing to take control of the game. Just after the hour mark, Antonio Valencia sent in a good cross which Pogba headed just wide.
Pogba then sent an audacious effort from the halfway line well over Adrian's crossbar to sum up United's lack of ideas in attacking areas.
The game seemed to be petering out before a flare up between the sides as Pogba's foul on West Ham skipper Mark Noble led to a fracas.