The England striker, selected in the absence of Robin van Persie, was netting for the first time in the Premier League since the opening day of the season and quickly added another when he swept home Antonio Valencia's cross.
"It is really important for Danny," said Moyes.
"He is a forward and he needs to get himself goals. He got two poacher types today and he could have had had another.
"With Robin out he stepped up to the mark. When you get opportunities, you show what you can do. Danny certainly did that."