The result left Chelsea one point behind the fourth-placed Hammers.
United almost had a penalty in the first-half after Callum Hudson-Odoi appeared to handle the ball.
But referee Stuart Attwell dismissed the appeal after consulting with the pitch-side monitor.
At the opposite end, Edouard Mendy was saved strikes from Scott McTominay and Fred.
Speaking after the game, Ole Gunnar Solskjaer was fuming about Atwell's penalty decision.
"It was a penalty, 100% nailed on," Solskjaer told BBC: If that's a natural position for a hand to be in when the ball comes in then I must be blind.
"We wanted to win. Apart from the last quality or pass being not good enough, we had great pressing, desire [and] work-rate. There was a great save by David de Gea and we got the clean sheet.
"That's what let us down, that quality on the ball. The first half was scrappy. Poor game really. But then we played some good stuff at times. When we settled we created a few chances but we didn't have enough."