In a game of precious few clear-cut chances, Luke Shaw hit the post and Paul Pogba headed wide with United's best opening.
A third blank in their last five league games for United means local rivals Manchester City can seal another Premier League record by winning the league by at least 19 points if they win at Southampton on the last day of the season on Sunday.
"We are the second best team in the Premier League, a fantastic competition with lots of good teams," said Mourinho. "We came second. I am not totally happy (because) that is not my nature, but since the moment we felt it was impossible to finish first, we had the target to finish second and it is done today."
In United's first game since legendary former manager Alex Ferguson had to have emergency surgery on a brain haemorrhage, the visiting fans showed their support to the 76-year-old Scot with persist chants of "we all love Alex Ferguson".
There was a more subdued atmosphere from the home fans at the end of another deeply frustrating campaign for the Hammers.
Two years ago to the day, West Ham waved goodbye to Upton Park in a thrilling 3-2 win over United on a night of high emotion in East London.
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.
Survival was only mathematically secured last weekend and West Ham remain in lowly 15th place.
"It's a big achievement for us," insisted West Ham manager David Moyes of keeping his side in the top flight. "We had to get the players in order a little bit after we came in, and we have done that. This is a big club and there are big expectations from the supporters so they will need to improve."
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