Moyes admits that he was readying himself for a replay as a fourth round tie entered stoppage-time with the scores locked at 1-1.
Everton had one last ace up their sleeve, though, and substitute Johnny Heitinga stepped off the bench to grab a dramatic winner.
"It was great to get through - the job is to get through in the FA Cup," said Moyes.
"We have had many years where we haven't got through in ties quite similar, so I'm pleased to get through.
"I thought it was going to a replay, I thought Bolton had done enough to earn a replay, but I thought if anybody was going to shade it and win the game then it was going to be Everton."
While pleased to pick up another win, Moyes did see Belgian winger Kevin Mirallas hobble off early on with what appeared to be a reoccurrence of his recent hamstring troubles.
Moyes added: "I don't think it's as bad as we thought, but you could see by how well he played and how well the team played in the opening 15 minutes or so that he's a really important player to us.
"He helps to knit a lot of things together and we missed him after that, we struggled for a spell leading up to half-time. I thought the second half we had more of the play without being creative and scoring goals. I thought we always looked the most likely to win."
Moyes saw Nikica Jelavic draw another blank at the Reebok Stadium and concedes that he would like to bring in a new frontman to help shoulder the goalscoring burden.
He said: "I would like to if I could, but I would be very much surprised if I did."