Tottenham Hotspur's win over Newcastle United on Wednesday night ensured that the Red Devils still required a point to guarantee runners-up spot behind local rivals Manchester City, and neither side could break the deadlock in the capital as Jose Mourinho's men moved four points clear of third-placed Spurs with one game to go.
The visitors came the closest to scoring when Luke Shaw hit the post in the first half, but the match ended as a stalemate as United recorded their first Premier League draw of 2018.
West Ham came into the match with the knowledge that they are guaranteed Premier League football again next season after results went their way last weekend, and they made the quicker start as manager David Moyes came up against one of his former clubs.
Marko Arnautovic has been the Hammers' most potent attacker this season, and he almost broke the deadlock after only seven minutes when a sharp turn and run created space for a 25-yard effort which forced David de Gea into his first piece of action.
Angelo Ogbonna was the next to threaten when he towered over Scott McTominay at the back post to nod a downward header into the arms of De Gea, but United did finally begin to settle into the game around the midway point of the half.
A mistake from Manuel Lanzini gifted the ball to McTominay in a dangerous position and his powerful effort flashed narrowly wide of the target, sparking a spell of intense pressure from the visitors.
Jesse Lingard's swerving drive needed to be turned away by Adrian moments later, but the Spaniard's best work would come in the 23rd minute when he produced a stunning double save to keep the scores goalless.
The first stop came from Alexis Sanchez as the Chilean swept Antonio Valencia's low ball towards goal and after parrying that out towards Shaw, Adrian then recovered in time to turn the full-back's powerful follow-up effort against the post.
West Ham weathered the storm, though, and were soon back on the front foot themselves with Cresswell firing a dipping effort just wide of the target, although De Gea looked to have it covered.
It was Arnautovic who continued to look most likely for the hosts, and he burst past Smalling with ease on a number of occasions in the first half - one of which saw him reach the byline before teeing up Joao Mario, who smuggled his effort wide of the near post from close range.
The Hammers had more than held their own in the first half, but United almost took the lead right on the stroke of half time when another swerving effort from Lingard had Adrian going in one direction before he quickly changed to parry the ball away.
Sanchez had endured a largely ineffective first half in which he dropped deep far too often, but he almost atoned for that within seconds of the restart when he linked up with Paul Pogba before beating Adrian to the ball inside the area. However, the angle was against him and his floated effort towards goal was comfortably nodded off the line by Ogbonna.
Lingard then stung the palms of Adrian from outside the area for a third time before Pogba met a Valencia cross shortly after the hour mark, only to put his downward header wide when he should have done better.
Smalling had a similar headed chance from a corner just four minutes later, but he too failed to make the most of it with a mistimed effort which sent the ball wide of the near post.
It was the visitors who continued to look the most like scoring in the second half, though, and Pogba came as close as anyone to breaking the deadlock when he collected Sanchez's pass before flashing a bouncing effort across goal and just wide of the far post from the corner of the area.
West Ham did hit back with a couple of ambitious sights of goal in the final 10 minutes, with Arnautovic slicing a first-time volley on the run wide of the target before turning provider for Noble, who was unable to replicate his goal-of-the-season contender against Leicester City as he skied his long-range strike well over.
The game looked to be petering out into a draw in the closing stages until Noble reacted furiously to a frustrated foul from Pogba, earning both players a yellow card - although Noble was perhaps fortunate his was not a different colour.
Lanzini had the final shot of the game with a curling free kick which De Gea watched all the way behind, and United showed no signs of pushing for the victory as Mourinho wound down the injury time by bringing on two defensive changes.
The result means that United have now failed to score in successive Premier League games for the first time since October 2016, and the Red Devils are still without a win in their final away game of the season since 2011-12 - now drawing five and losing one of their six since the dramatic finale which saw Man City pip them to the title.
However, the clean sheet does ensure that De Gea will claim the Golden Glove award in the top flight this season, while it also wraps up United's best finish since the retirement of Sir Alex Ferguson.
The Red Devils will end their Premier League season at home to Watford on Sunday before taking on Chelsea in the FA Cup final on May 19, whereas West Ham will bring their tumultuous campaign to a close at home to Everton.
WEST HAM (3-4-3): Adrian; Rice, Ogbonna, Cresswell; Zabaleta, Kouyate, Noble, Masuaku (Carroll 66'); Mario, Arnautovic, Lanzini
MAN UTD (3-5-2): De Gea; Lindelof, Smalling, Jones; Valencia (Bailly 94'), McTominay, Herrera, Pogba, Shaw; Lingard (Rashford 74'), Sanchez (Young 92')