Manager Mauricio Pochettino made six changes for the fifth round clash at the Stadium of Light, with Craig Gardner's goal five minutes after the break sending the League Cup finalists through.
Southampton goalkeeper Artur Boruc and defender Jose Fonte did not see any action, while Jay Rodriguez and Morgan Schneiderlin were only used as second-half substitutes.
The damage had been done by then, as Southampton could not find an equaliser, with Rickie Lambert guilty of shooting over from point-blank range in the latter stages.
Despite the defeat, Perez believes it was the right decision to rest players as they seek to improve on their Premier League position of eighth.
"We played the best team we decide to play," he said in quotes reported by The Daily Echo.
"We assess our players, we know how they are, we know which problems each other got from the last game and we decide the best team to play.
"I repeat again we trust every single player we have in our squad."
Sunderland have now sent Southampton out of both the FA Cup and the League Cup this season, with the Premier League left as Southampton's only focus.
Perez claimed the size of their squad had impacted on their progress in the competition.
"Our squad (does) not (have) depth in this moment so we have a lot of young people from the under-21s so we are not disappointed," he continued.
"(We are) just a little sad to be out of the FA Cup because we tried to move forward."
