Pochettino's men appeared well-placed to stretch their unbeaten run in the Premier League to seven matches when Harry Kane extended his rich vein of form with a well-taken 49th-minute opener.
But Gayle levelled matters in the 69th minute after Benjamin Stambouli lunged in on Joe Ledley and Palace provided a rousing finish for a vocal Selhurst Park crowd.
Wilfried Zaha crowned a fine substitute cameo by settling up Jason Puncheon to ensure Alan Pardew's first league game in the Palace hotseat ended in triumph.
"Maybe when we scored for 1-0 I think that we need to push a little bit more," Pochettino said.
"But in the end it is clear that the penalty changed the game.
"Maybe it wasn't a penalty, maybe it was – I don't know, we need to analyse on the TV.
"But this was the key moment that changed the energy in the stadium and on the pitch.
"After we suffered a little bit and Crystal Palace believed they could get the three points.
"There was a lot of long ball and today, in this challenge, they won more than us.
"Maybe Crystal Palace pushed more than us, were more fresh than us – we have to analyse."
Tottenham's recent impressive run had helped them bridge the gap to the Premier League's top four.
Victory would have taken them above Southampton and into fourth place ahead of the latter's trip to Manchester United on Sunday and Pochettino conceded the Palace reverse represented an opportunity missed.
"We are not happy, we are disappointed with the result," he added. "We missed a good opportunity to stay [in touch] with the top four."
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