The Sky Bet League One club produced a tenacious and well-organised display to frustrate big-spending City, who boasted three debutants in centre-backs Jens Hegeler and Bailey Wright, plus Bosnia international striker Milan Djuric, in a goalless draw at Ashton Gate.
Rosler said: "Keeping our good record going against a Championship club who have just made three great signings was very pleasing.
"They controlled the game in the second half, but it was credit to our defensive organisation and attitude that we restricted them mainly to shots from outside the box.
"It's fantastic for the club to be in the fourth-round draw for the first time. But what I embrace most is that my players stepped up to the tempo of facing opponents from a higher league.
"Their new players gave Bristol City a lift and they played well, which makes our performance all the more satisfying.
"It was a fair result and a great one for us. We need the money more than Bristol City!"
Fleetwood deserved their replay on Tuesday week, having carved out the best chance of the game after only eight minutes.
Devante Cole's low cross was met at the near post by David Ball and City goalkeeper Frank Fielding spread himself to save.
City lost Fielding and captain Korey Smith to first-half injuries, but had the lion's share of possession throughout.
Mark Little forced a good first-half save from Alex Cairns, while Djuric shot into the side-netting when well placed.
The second half saw the hosts become increasingly frustrated as their measured attacks lacked a final penetrating pass.
Tammy Abraham went close on several occasions, but Fleetwood stood firm.
City head coach Lee Johnson refused to be downbeat, despite seeing his side fail to win for the seventh successive game.
He said: "I was quite pleased with the way we played. The new boys were fantastic and we had just one day to work with them on a new formation.
"Jens Hegeler looked like a Rolls Royce, while Bailey Wright did what it said on the tin with his performance and big Milan Djuric also did well.
"It was almost like a chess game at times and, while I am disappointed not to have won, I am not going to be defeatist.
"We stopped the rot of defeats, kept a clean sheet and dominated possession. So we can move forward to our next Championship game against Cardiff with a bedrock of a solid display.
"We need to get the ball forward quicker and we will work on that."
Bristol City last week