The home side made 11 changes to their team, with Adam Lallana and Jay Rodriguez the only two starters from Saturday's win at Liverpool included in the 18-man squad.
Despite the alterations, Mauricio Pochettino's side booked their place in the fourth round with Gaston Ramirez and Jos Hooiveld on target.
But the visiting Robins, who have so far failed to win in League One, produced an impressive performance against their Premier League opponents.
And Perez was quick to praise the opposition after the game, saying: "We were obviously made to work hard in the second half. I thought the game was difficult.
"We scored in the first half but the opponent played well. They played without pressure.
"We are very happy about the result because it became a hard training session towards the end of the game.
"We know it is difficult to play as a team when you've made 11 changes, but this type of game, the cup games, we have to fix these kind of problems.
"We know it is not for the players playing the best football.
"I think Bristol played well and it was not easy for us to get the ball back and play with possession.
"At the end of the game, when Harrison Reed and Jay Rodriguez came in it helped the team create the level and tempo.
"It was because the opponent played well, not because we dropped off that they were better in the second half. They got some chances to score so it was difficult."
Joe Bryan, Jay Emmanuel-Thomas, Bobby Reid and Stephen McLaughlin all threatened to level for the west country outfit, who dominated for large spells of the second period.
It was a performance that belied their lowly league position and one which manager Sean O'Driscoll enjoyed.
"I am pleased with the way the players performed, pleased with the way we approached the game," he said.
"We got stronger as the game went on, which is pleasing.
"We came to terms with their possession, which you expect when you play a top team."