Marouane Fellaini handed the hosts a 1-0 half-time lead, but it was their two wide men who provided the most telling performances.
Pienaar scored a sublime second goal with a 20-yard rocket into the top corner and then brilliantly set up Mirallas to net a clinical third.
Moyes said: "Steven's was a really good goal coming in off the right, and Kevin on the other side looks to be finding form, which we need.
"We need the people who play in those positions to get us the goals. [Nikica] Jelavic is getting closer to scoring, but Kevin Mirallas, in the last couple of games, has got us important goals.
"It has been a little bit tense, but the players put that to bed today. They scored three and could have scored another three."
Victory was Everton's first in four attempts and has reignited their bid for European qualification.
Moyes believes that with minor improvements, his side can now go on to finish the season strongly.
He added: "Overall, I'm pleased with how they're going about their work. The players have been playing quite well. They are doing a good job.
"The players have to keep going and we just have to hope we get goals to go with the chances we are making and when we are put under pressure we keep a clean sheet."