Mourinho was dismissed midway through the first half after reacting poorly to the referee's decision to book Paul Pogba for diving.
Manchester United once again flattered to deceive as they struggled to break down the Hammers' well-drilled defence.
The visitors took the lead after only two minutes when Dimitri Payet whipped in a free-kick that found Diafra Sakho who was able to get in front of Zlatan Ibrahimovic and head home from six yards out.
The home side struggled to respond and Pogba lashed an ambitious long-range effort high and wide before captain Juan Mata saw his shot gathered by Darren Randolph.
Mourinho's side did finally equalise in the 21st minute though when Pogba played in Ibrahimovic ,who flicked his header over Randolph.
After Mourinho had been sent off, Jesse Lingard crashed an acrobatic effort off target and Randolph kept out Marcus Rashford's drilled shot.
On the stroke of half time Angelo Ogbonna did superbly to stop Ibrahimovic before Randolph saved Lingard's fierce drive.
West Ham started the second half well and Sakho went close to restoring their lead but he could not get his header on target before David De Gea turned Payet's free-kick over the bar.
Manchester United struggled to find any cohesion in the second period although, after Wayne Rooney had come on as a 64th-minute substitute, the England international fired in a weak effort that Randolph saved before the Hammers keeper pushed Lingard's shot on to the post.
The impressive Randolph then denied Pogba and Ibrahimovic in the closing stages but Slaven Bilic's side almost grabbed a dramatic winner when substitute Ashley Fletcher went close to beating De Gea.