The League One club had taken the lead through Cole Stockton's cool finish only for Jermaine Beckford to level for the Championship side just after the hour mark, as the game eventually required penalties to separate the teams.
Owain Fon Williams was the hosts' hero in the shoot-out, saving Andre Moritz's and Alex Baptiste's spot-kicks to allow James Rowe to score the winning kick and Moore could not hide his delight at the result.
"It was a great cup game, I'm proud of them," he said after the game.
"Sometimes when you concede two disallowed goals, sometimes you think is it going to be your night.
"They've held the line well and there was a fantastic clearance from Evan Horwood off the line, so we were hanging on a little bit in the second half, but we're playing Bolton who are full of class and quality players.
"I think the half-time lead was deserved. We would have been disappointed to come in without it and it was a fantastic goal as well for Cole.
"It was an accomplished striker's finish from Cole as normally it's boom time for him but he's going to mature into a great striker.
"I knew they were going to come out all guns blazing, which they did with four up top.
"It's difficult when you look at the quality they've got - it's daunting - but we dug deep and held in there and got the breaks where on another day Bolton might have got them.
"We'll take what we can get - if Liverpool or Everton get through and anything here, we know we've got a chance."
In contrast, Bolton boss Dougie Freedman said he felt that his side deserved to have progressed and that losing the game was difficult to take.
"I felt that we'd done enough to win the game," Freedman said.
"Our decision-making around their box went wrong though and let us down.
"We had enough opportunities up and around their box - including a couple of goals disallowed for offside - so our decision-making was the key that let us down.
"I'm disappointed to lose the match but I felt we had done enough - our approach play was good. But good luck to them.
"It's very difficult to tell about Tom Eaves' shot off the line. A couple of goals were disallowed and it's difficult as unless you hit the back of the net, you don't know. I would maybe just put it down to our decision-making being better in front of goal."