An injury-time winner from John Heitinga gave the Toffees a 2-1 win over Bolton Wanderers in their fourth-round clash.
Moyes admitted it was "great" to make it through and claimed that, although Bolton worked hard, his side deserved to "shade the game".
"It was great to get through. The job's to get through in the FA Cup and I've had many years where I've not got through in ties quite similar so I'm pleased to get through," said Moyes.
"I thought Bolton had done enough to get a replay but I thought if anyone had done enough to shade the game it was Everton.
"We needed to try another tactical way to create a goal. We played [Marouane] Fellaini a bit deeper today and for the last 15 minutes I thought I might be worth pushing him further forward to see if we could get anything. It wasn't him who got it, but it was another player."
Everton are currently sitting in fifth place in the Premier League table.