Three early goals from Antonio Valencia, Robin van Persie and Wayne Rooney gave the Premier League champions a comfortable cushion at Craven Cottage and United eventually ran out easy winners, despite Alex Kacaniklic's second-half reply.
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"Jonny's back seized up; Rafael had gone over on his ankle and Tom Cleverley had double vision," Moyes told Sky Sports. "We actually had four [injured players] - Patrice Evra had an injury as well.
"It never helps the continuity. We didn't play as well in the second half but nevertheless we were always worthy winners.
"All credit to Fulham - they kept going and got their goal."
Moyes was delighted by United's start to the game, which saw Valencia, Van Persie and Wayne Rooney score inside the opening 22 minutes.
"The first 20 minutes we were outstanding – great goals, great movement for the goals," added the Scot.
"We knew it would be tough, because Fulham were off a bad result last week [a 2-0 defeat against Southampton]. But we got the goals and we played very well."