Former Republic of Ireland winger Damien Duff, who retired from international football on Friday, scored his second goal of the season to give Fulham a third-minute lead.
But United’s £24m signing Van Persie equalised before Shinji Kagawa and Rafael both netted to leave Sir Alex Ferguson’s side 3-1 ahead at half-time.
David De Gea failed to collect a cross in the 64th minute which led to United captain Nemanja Vidic inadvertently turning the ball into his own net, but the home side hung on to claim three points.
“It is always good on your debut to score a goal like that. I enjoyed it,” Van Persie told BBC Sport.
“Fulham are a good side, they like to play football and it is always hard against them. In the end we got the three points and that is the main thing.
“Old Trafford is a big stadium, with great fans and they have been very nice to me today, so I really appreciate it. This is a new challenge that I am looking forward to.”
Meanwhile, Ferguson admitted United were inconsistent against Fulham, with the Red Devils manager particularly frustrated by the visitors’ second goal.
“I think we saw the two sides of Manchester United,” Ferguson told MUTV.
“After cruise control, we made a terrible mistake for the second goal and it became a match. It should never have been a match, some of our football was fantastic. But we lost that second goal when we should have been in easy street.
“Fulham tried to slow the tempo of the match down. They do it very well because they’ve got several players over 30 years of age, they’re a very experienced team. After their second goal, we struggled.”
Wayne Rooney faces four weeks on the sidelines after the striker was carried off on a stretcher with a bad gash on his knee.
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