Two goals in the opening quarter hour – from Mark Byrne and Olly Lee – put the visitors on the back foot and they never recovered as Gillingham sealed a comfortable passage into the third-round.
“Make no bones about that, it was a disappointment,” said Moore.
“We thank the travelling fans that have made the long journey down here. We send our apologies to them because that’s certainly not what we pride ourselves on in terms of here at the football club.
“When you come down to places like this you’ve got to be ready.
“I’ve been here a number of years as a player myself and I’ve been here as a coach and it’s a certain type of ground that you have to be mentally strong down here and physically right to deal with their threat and I thought today we were lacking in those areas.”
Byrne opening the scoring in style after 11 minutes with his first goal of the season, firing past Seny Dieng after controlling the ball brilliantly with his chest.
The Gills’ second was even better as Lee latched on to Tom O’Connor’s lofted pass and lobbed a hapless Dieng from 19 yards.
O’Connor somehow squandered a golden chance to put Steve Evans’s side 3-0 up within 25 minutes when he shot wide from Brandon Hanlan’s clever cutback.
Rovers, who sit two places above the Gills in League One, never looked like matching the feat of reaching the fifth round last season.
Rakish Bingham headed wide with their best chance of the opening half, while James Coppinger shot straight at Gills goalkeeper Jack Bonham.
Hanlan sealed Gillingham’s victory after 68 minutes.
Moore said: “We just didn’t quite do our jobs today and I know, the players know, we’re a tight knit group who pride ourselves on that so it’s not about throwing our toys out of the pram, so to speak, because it has been the first “below” performance of the season.
“What we’ve clearly identified is that a performance like that cannot happen for the remainder of the season.”
Gillingham: Bonham, Tucker, Ehmer, Ogilvie, Fuller (Hodson 90), Jones, Byrne, O’Connor, Lee, Hanlan (Willock 90), Mandron (Ndjoli 90). Unused substitutes: Charles-Cook, Marshall, Walsh, Jakubiak.
Doncaster: Timothy Dieng, Halliday, Wright, John, Reece James, Whiteman, Sheaf, Sadlier (May 64), Coppinger, Gomes (Thomas 64), Bingham (Taylor 84). Unused substitutes: Lawlor, Amos, Blaney, Greaves.
Referee: D Handley (Lancashire).