The Toffees went down to an Adam Johnson goal on the stroke of half-time, but fought back with goals from Nikica Jelavic and Marouane Fellaini.
Moyes was relieved that his side got the win to remain in the top four.
"I thought it was going to be one of those days," said Moyes. "[A goal] didn't look as if it was coming.
"I thought we played quite well in the first half but found ourselves a goal down, and fortunate not to be two down.
"We didn't deserve that with our possession. We kept going and the players showed tremendous character to come from a goal behind. Maybe our pressure told a little in the end."
The win was Everton's first in five and keeps them in the Champions League places.