Spurs endured a nightmare start at Wembley as Willian fired home an early free-kick but soon levelled through Harry Kane's stooping header.
Willian added another from the penalty spot and Dele Alli responded after the break to draw the north Londoners level again.
Substitute Eden Hazard put the Blues in front for a third time before Nemanja Matic wrapped up victory with a long-range effort late on, consigning Spurs to a seventh FA Cup semi-final loss.
But Pochettino is adamant his players should be praised for the way they performed at England's national stadium, despite failing to get the better of their rivals.
"4-2 looks like a big result for Chelsea but it's a game I feel proud of," he insisted.
"We gave all that we had. It's true we lost but it's the way we wanted to play.
"We respected our philosophy. The team showed great personality and character.
"If you see the game without emotion you will see that Tottenham played much better but they were clinical.
"We are disappointed but now we move on and try to be calm.
"We forced them to play deeper and play the counter attack.
"Congratulations to them and we will try to fight for the Premier League."
Spurs have little time to dwell on their Wembley disappointment with a trip to Crystal Palace to come on Wednesday night.
The north Londoners are just four points adrift of Premier League pacesetters Chelsea and Pochettino is convinced his players will suffer no hangover from their defeat to Antonio Conte's men.