But in keeping with his up-and-down campaign, Rooney then fired a 78th-minute penalty over the crossbar following a handball by Jordan Spence.
It was United's fifth failed penalty of the season and Rooney's second. After missing against Arsenal, Alex Ferguson warned that he was in the last-chance saloon with his penalty duties.
Judging by Ferguson's exasperated reaction to this one, Rooney will not be taking any more for a while.
Such is the depth of United's squad, Ferguson was able to make 10 changes to his starting line-up from Sunday's victory against Liverpool - full-back Rafael the only player deemed not to require a rest - with the manager raiding the treatment room for three of his team.
Rooney, Nani and Anderson, all of whom have been sidelined due to injury, returned to action. But there was not even a place on the bench for the 'concussed' captain Nemanja Vidic or midfielder Darren Fletcher, whose comeback from a bowel condition has stalled since the Christmas period.
United did not appear hampered by the absence of many of their first-choice players, however, with Anderson in particular making an impressive contribution in the early stages.
Making his first appearance since injuring a hamstring against Reading on December 1, he dominated the central area and instigated the move for Rooney's ninth-minute opener.
Anderson's sublime pass from inside the United half found Javier Hernandez 30 yards from goal, but the Mexican's first touch enabled him to cut into the penalty area before he unselfishly squared the ball for the unmarked Rooney to slide in from close range.
United assistant manager Mike Phelan was caught on camera mouthing "great pass by Anderson" to Ferguson.
Rooney, lacking sharpness after his three-week lay-off, was content to play a deep-lying role behind Hernandez, but United lacked a cutting edge and had to wait until the 28th minute before creating their next chance.
It required a crucial clearance by West Ham youngster Daniel Potts to deny United a second, though, with the full-back sliding in at the far post to keep out Nani's goal-bound half-volley.
Having twice taken the lead through James Collins in the first game at Upton Park, which finished 2-2 thanks to Robin van Persie's stoppage time equaliser, there was a sense that West Ham had blown their best chance of progression as they toiled away in the first half of this replay.
Sam Allardyce's team were flat and unambitious, with forwards Carlton Cole and Ricardo Vaz Te isolated by a lack of service, but the Old Trafford pitch was little help either.
Ferguson claimed last month that the incessant rain throughout December had seen the pitch "collapse", forcing him to insist on the surface being relaid at the end of the season, and it did little to encourage either team to pass the ball.
The unpredictable bounce of the pitch probably contributed to the mistake by West Ham defender Alou Diarra which led to the first chance of the second half for the home side.
Diarra's failure to control the ball led to Hernandez being gifted a free strike on goal from 18 yards, but just as the centre-half had been compromised by the pitch, the United forward blazed his shot over the crossbar when a truer surface would surely have ensured a strike on target.
Having survived that scare, West Ham grew into the game.
The nerves were jangling, and Rooney could have eased them by scoring his penalty. He missed again, but, on this occasion, it did not prove costly.