The England international scored on his comeback from injury but also missed a late penalty as the Red Devils booked their place in the next round with an unconvincing win
Sir Alex Ferguson made 10 changes for the game, with Rafael the sole man retained from the weekend win over Liverpool, and with Wayne Rooney and Nani making their returns from injury.
West Ham, meanwhile, made six changes with Joe Cole, Kevin Nolan, Matt Jarvis and James Collins, the man who got both goals at Upton Park in the original tie, all missing out.
The hosts began quickly and took just nine minutes to open the scoring, with Rooney the man to break the deadlock with his ninth goal of the season.
The goal came from a quick break, Anderson’s superb through ball releasing Chicharito down the left. The Mexican scampered into the area before unselfishly squaring the ball to Rooney to tap home from close range.
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United might have doubled their lead just before the half-hour mark when they twice went close. A short corner saw Ryan Giggs and Antonio Valencia work an opening for Nani who smashed the ball across goal, beating Jussi Jaaskelainen but not Daniel Potts who hacked the ball to safety just inches from his own goalline.
Moments later, Chris Smalling almost grabbed a second but the defender could only plant his header straight at Jaaskelainen when well placed.
A more threatening Hammers side emerged for the second half, dominating the play, yet it was the hosts who again had the best chance of the opening exchanges.
Rafael survived a handball appeal in his own area to bring the ball out of defence before trying to pick out Chicharito. His pass should have been dealt with by Alou Diarra but a mistake by the West Ham man left the striker through on goal, yet he could only blaze wildly over the bar.
West Ham started to pile the pressure on and came within inches of grabbing an equaliser on the 60 minute mark. A long ball from Jordan Spence found Matt Taylor in space, he rolled the ball across goal trying to find Ricardo Vaz Te at the far post, with United needing Rafael to smash it clear inches from his own goal.
Back came United, with Rafael pouncing on a loose ball before poking it towards Chicharito, but Diarra just got enough off a touch on the pass to allow Jaaskelainen to claim before the striker.
The chances continued to come in a tense finale with Smalling denying substitute Kevin Nolan before Giggs' cross was handled by Spence in the area only for Rooney to blaze the spot kick high over the bar.
Ultimately, however, Rooney's early effort was enough for United who continue to fight on three fronts and head into the fourth round and a visit from Fulham.