The Old Trafford captain became the third highest goalscorer in Premier League history when he broke the deadlock in the fifth minute of Saturday's clash, but his red card just before the hour for lashing out at Stewart Downing put his team under unnecessary pressure.
Rooney's early strike saw him move onto 176 goals in the top flight and overtake Thierry Henry, with only Alan Shearer (260) and Andy Cole (187) now ahead of him.
Robin van Persie extended the hosts' lead after 22 minutes but it was halved before the break as Diafra Sakho took advantage of some more poor defending from Manchester United.
Rooney's dismissal enabled the visitors to push forward in the final 30 minutes, and they had a goal disallowed late on when substitute Kevin Nolan was adjudged offside, but Manchester United were able to hold on to claim a second win under Louis van Gaal.
With injuries ravaging his squad, the Dutchman was forced into two changes in defence, with Paddy McNair and Luke Shaw coming in for their debuts in place of the injured Jonny Evans and the suspended Tyler Blackett.
West Ham also made two alterations from their 3-1 win over Liverpool, as Morgan Amalfitano and Diego Poyet were handed their first starts, but the visitors were not given time to settle and Rooney opened the scoring after five minutes.
Rafael da Silva was allowed to race clear on the right and, with no defender closing him down, the full-back perfectly found his captain, who applied a first-time finish to beat Adrian from 12 yards.
Midway through the half, after Enner Valencia and James Tomkins had wasted good chances for an equaliser, the visitors found themselves two goals behind.
When Alex Song was caught in possession inside his own half, the loose ball dropped to Falcao. He found Van Persie and the Dutchman rifled the ball into the bottom corner.
However, despite their two-goal cushion, the hosts came under pressure late in the half and West Ham were rewarded with Sakho's third Premier League goal of the season in the 37th minute.
David de Gea, making his 100th league appearance for United, failed to claim a right-wing corner and although Valencia's header hit the crossbar, Sakho was on hand to head home the rebound.
Sakho had a chance to double his tally nine minutes after the break as a deep cross from the left found him at the back post, but his volley was pushed away by De Gea.
As the hour approached, the game was turned on its head as Rooney allowed his frustration to get the better of him.
Downing looked to mount a counter-attack from a poor Manchester United free-kick but the midfielder was stopped when Rooney lashed, an act that resulted in referee Lee Mason issuing a straight red card.
Wave after wave of West Ham attacks followed, and the visitors thought they had rescued a point when substitute Nolan met Carl Jenkinson's cross to convert from close range with two minutes remaining, but they were denied by the assistant referee's flag.
