The Red Devils could not find a way to beat Adrian, but Jose Mourinho's Manchester United will finish second in the Premier League.
Manchester United guaranteed a second-place finish in the Premier League with a tame 0-0 draw at already-safe West Ham on Thursday.
Jose Mourinho's men, beaten 1-0 at Brighton and Hove Albion on Friday, knew a point at London Stadium would be enough to ensure they will end the season as runners-up to rivals Manchester City.
With defences on top, Adrian made a string of saves to protect West Ham's clean sheet and justify his selection in goal ahead of England international Joe Hart, who returned from illness but was only named on the bench.