Danny Welbeck scored his first goal at Old Trafford since October 2012 before Adnan Januzaj made a mug of James Collins to double the advantage prior to the interval.
Ashley Young netted another brilliant effort in the second half so Carlton Cole's consolation did not threaten a fourth successive win, equaling the best run of Moyes' short reign so far.
"The rhythm in attack is beginning to get better," said Moyes.
"We are beginning to create more chances. We had the chances to score maybe four or five but overall we were really pleased with how we played."
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