The former Olympiacos man finished as top scorer in the Greek league last season with 20 goals, and looked sharp cutting in from the right flank during the clash at Goodison Park.
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His lively performance did not escape his manager's attention, and Moyes is full of praise for the Belgian international.
"He had three or four opportunities that he made himself," the boss told evertonfc.com. "I thought he played well. He'll give us another attacking player who looks like he might have a goal in him.
"He can play either side for us and he'll play through the middle as well.
"He scored a lot of goals in Greece and I think you could see tonight, every time he got the ball you thought he was going to try and get a goal and try to shoot somehow.
"Hopefully he does that and a few go in for us."
Mirallas could play a more central role in the coming weeks as Nikica Jelavic faces a spell on the sidelines through injury, but Moyes also highlighted the impact of Victor Anichebe, who replaced the Croatian international against the Magpies and scored his side's second goal.
"He got the winner that went over the line which they didn't give," added Moyes. "You've got to say, on the night, he had two goals - he did score one and he did score another that wasn't given."