Postecoglou praises Tottenham's new-found mental strength
Spurs closed to within two points of fourth-placed Aston Villa who were playing later and opened a six-point cushion over Manchester United, who face the daunting trip to Manchester City on Sunday, in the fight for a place in next season's Champions League.
A top-four finish would guarantee Postecoglou's men a return to Europe's elite competition, while fifth could be enough depending on the success of English sides in the remainder of this season's European competitions.
Eberechi Eze's free-kick had fired Palace into a shock lead on the hour mark.
But for the third time in their last four home league games, Spurs came back to win despite conceding first.
Timo Werner's first goal for the club started the fightback 13 minutes from time.
Cristian Romero headed in three minutes later before Son Heung-min sealed the points on the counter-attack.
Spurs have been criticised for a lack of mental strength in the past and Postecoglou believes his side are showing that is no longer the case.
"It was a game we should have won a lot more comfortably but we had real belief in the process and were not getting swayed by things not going our way. I thought the mentality was first rate," said the Australian.
"You need that goal to break open a team that is going to sit so deep. I still felt like we were putting enough work into them that at some point we would be able to break them.
"Obviously we conceded which was disappointing. You're looking for a reaction and I thought the reaction was outstanding.
"They just had a real belief today in our processes and our football and I'm really pleased with the outcome."
Werner was guilty of missing a glorious chance to open the scoring in the first half but made amends when he tapped home Brennan Johnson's cross.
Postecoglou is hoping the German will be infused with confidence after breaking his duck for the club in his seventh appearance.
"For all of the attacking players goals are important. I thought he was having good game generally," added Postecoglou.
"He didn't stop believing, got into the right area and outstanding work from Brennan."