Falcao had a great opportunity to score his first goal for United early on but the Colombian could not quite reach Rooney's free kick and the chance went begging.
© Getty Images
The pace of Angel Di Maria was a dangerous tool for United and the Argentine teed up Falcao for another chance but he could only put an effort well wide from 25 yards out.
United grew into the game as the half went on, and really should have had a third through Di Maria as he connected with Ander Herrera's cutback. Unfortunately for the hosts, his shot flew harmlessly wide of the target.
The missed chance seemed to galvanise West Ham, and they pulled a goal back through Diafra Sakho, who was all alone to nod in from close range after Reid hit the post from a corner.
© Getty Images
Following the restart, Falcao again came agonisingly close to opening his United account with a deflected effort but Adrian showed great reflexes to tip away the ball at the last moment.
For all their attacking prowess, however, United's problems in defence continued and Sakho was unlucky not to equalise for the visitors after getting onto the end of Aaron Cresswell's cross.
A huge moment in the match then saw Rooney sent off after the United captain challenged Stewart Downing high up on the thigh to stop a West Ham counter-attack.
From then on the Hammers controlled the game, with United struggling to get a hold of the ball.
The London club thought they had grabbed a winner in the dying minutes but Kevin Nolan's close-range finish was disallowed for offside, much to the relief of the Old Trafford faithful.