The Toffees took the lead in the first half at the Reebok Stadium before being pegged back by Marvin Sordell and had to withstand plenty of pressure before Johnny Heitinga's late strike.
And, despite admitting that the nature of the winning goal was perhaps fortunate, the Bury-born midfielder expressed delight at his side's battling qualities.
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"We set off really well and for the first 15 or 20 minutes we looked as though we'd found our rhythm. The goal came at the end of a period of total domination and I thought it was going to be a quite simple game to win.
"Then, obviously, we conceded a goal and after that Bolton put us under a lot of pressure and we didn't play well enough after that.
"It was a proper cup tie and sometimes in a cup tie you just need to make sure you're in the hat for the next round. And that's what we are."
Neville went on to insist that nothing less than nine points from the upcoming league meetings with West Brom, Aston Villa and Manchester United will do.
He added: “It's a big week. We're looking for three victories and we've started it well.”