Saints made 11 changes to their side at a quarter-full St Mary's, 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, it looked like it would be an easy third-round win for Mauricio Pochettino's side after Gaston Ramirez rifled home a sumptuous volley from the edge of the box after 15 minutes.
However, while they have so far failed to win in League One, City knocked out Crystal Palace in the previous round and produced a similarly impressive performance in the second period.
It took Jos Hooiveld's late header to secure safe progression and a 2-0 win, with assistant manager Perez quick to praise tonight's opposition after the game.
"We were obviously made hard to work in the second half," said the Spaniard, speaking instead of manager Mauricio Pochettino. "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."
While keen on winning the competition, Saints are using the Capital One Cup as a way of giving their squad players a run out.
One of the 11 brought in was Tadanari Lee, returning to the starting line-up just weeks after his Saints career looked to be over.
Barely used last term, he was farmed out to FC Tokyo on loan for the second half of the season and told he had no future at St Mary's upon his return.
However, Pochettino had a change of heart after being impressed by Lee's attitude and aptitude, leading to his return to the side this evening.
"He played very well," Perez said. "We were talking on the bench about his performance, especially in the first half.
"He was very dynamic, he played not only as a forward but also tried to help team link up and build between the lines.
"I am really, really happy with his performance. It is good news."
Lee, like many other Saints player, failed to really shine during a second half in which City impressed.
Joe Bryan, Jay Emmanuel-Thomas, Bobby Reid and Stephen McLaughlin all threatened to level for the Robins, 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."