The Toffees were not at their best at the Reebok Stadium on Saturday afternoon but dug in before finding a last-minute winner through substitute Johnny Heitinga
The Toffees took an early lead at the Reebok Stadium through Steven Pienaar but were dragged back by Marvin Sordell's strike before half-time.
But, after seeing substitute Johnny Heitinga fire home to seal a place in the fifth round, the Scot insisted 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," he told reporters.
"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 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."
Belgian winger Kevin Mirallas' return to the startling line-up was cut short by a recurrence of his hamstring problem, but Moyes believes the blow may not be as severe as first feared.
"I don’t think it’s as bad as we thought but obviously you could see by how well he played and how well the team played in the opening 15 minutes that he’s a really important player to us and helps knit a lot of things together," he added.
"We missed him after that; we struggled for quite a spell. That 20-30 minutes leading up to half-time we just never really got flowing after that.
"I thought in the second half we had most of the play and, without maybe being that creative and scoring goals, I thought we always looked the most likely to."
The Scot also conceded that he may rue his decision to name a full-strength side with a midweek meeting with West Brom approaching.
"It might be good for you to say ‘David Moyes is putting his best team out’ but I might run aground later on because my players are running out of legs and that’s why managers rest their players," he said.
"There were periods today where I was looking over my shoulder saying ‘what can I do to make this better or change it?’ and I didn’t have loads of options.
"It’s a big week but if I’d come here with a weaker team and lost the questions would have been completely different."
Nikica Jelavic continued his barren run in front of goal against the Trotters, a fact which the 49-year-old went on to admit is a concern.
"I said I want to play him, I want to give him every opportunity to get it right. I felt that today might have been an opportunity for him and I’ll need to look at it now," he continued.
"Obviously I want to give him every positive I can and show him that I trust him to play and score."
Moyes' opposite number Dougie Freedman, meanwhile, was effusive in his praise for the Bolton staff who had helped clear snow in order to allow the game to ahead.
The former Crystal Palace boss also insisted that there were plenty of positives to take from his side's last-gasp defeat.
"I’m disappointed in the result, but not the performance. I felt that for the first 20 minutes we started nervously, I can’t put my finger on why but we did," he said.
"They got a goal up – a deserved goal, no complaints – but I thought the way we handled it, the way we got on with it, it didn’t set us back.
"For the next hour we played possibly the best football since I’ve been here, got a deserved goal and then tried for that winner, pushed everything forward and couldn’t quite get it.
"A small detail at the end has let us down; if you concentrate for 95 minutes against a Premier League side they’re going to punish you."