Southampton dominated the first half and went in at the break with a two-goal advantage thanks to superb strikes from Nathaniel Clyne and Rickie Lambert.
But Burnley responded well after the break, with goals from Sam Vokes and Danny Ings levelling matters only for ex-Claret Jay Rodriguez to step off the bench and put Saints back in front.
Adam Lallana extended the advantage only for Kevin Long’s late header to set up a tense finale, but Southampton held on to progress through to round four.
“We say in these type of games that anything is possible because the teams are playing without pressure,” Pérez told Saints Player
after the game.
“We started well and went 2-0 up but then didn’t do so well and they equalised which woke us up. Then we tried to play our way again and scored another two goals.
“They had good chances, so it was a good game and like I said before, they are a good team in the Championship,” added Mauricio Pochettino’s number two.
“They are doing well and have clear style and got chances because they deserved it.
“The most important thing is to get through to the next round because this was a difficult game coming at the end of a tough run for us.
“The players have maybe been showing some psychological fatigue so we are pleased that we are through and don’t have to replay the game again like some other teams will have to.”
Pochettino made five changes to the team that was beaten by Chelsea on New Year’s Day, but still selected a strong team in a competition that a number of Premier League managers use to rotate their squads.
“We always said that we would put out the best team to play the game,” reasoned Pérez, who confirmed that goalscorer Lambert was substituted after suffering a dead leg in the first half.
“We assessed the players yesterday after training and decided to pick this first eleven because it was the team we wanted to face Burnley.
“Sometimes there are strange situations like today when there are a lot of goals in a game that isn’t particularly clinical on our part, but as I have said, the most important thing is to stay in the draw.”