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Pochettino -- whose side lost ground on the leaders Manchester City after losing 4-2 to their North London rivals -- said on Tuesday it is logical to back one's player in such circumstances.
Dier, 24, celebrated bringing the game level at 1-1 by putting his finger to his lips, with Arsenal substitute Stephan Lichtsteiner reacting angrily to the gesture.
Referee Mike Dean showed the England midfielder a yellow card after Pochettino sprinted down the touchline to play peacemaker.
"To be honest, I didn't talk with him," Pochettino said at his press conference on the eve of their league clash with his former club Southampton.
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"Because when you do something and then it's so evident what happened after, you can take it in one of two directions.
"If you're a Tottenham fan, you take it in one direction, if you're an Arsenal fan, you take it in another direction.
"If you're the manager of Tottenham or Arsenal, or the Arsenal players doing the warm-up, you're going to back your own players."
Both clubs were charged by the Football Association on Monday with failing to ensure their players conducted themselves in an orderly fashion during the melee that followed Dier's goal.
Pochettino, whose side are fifth and eight points adrift of Manchester City, said there was little point in arguing over the rights and wrongs of Dier's actions.
"There's no sense now to talk too much, now it's just to accept the FA charge or not," said Pochettino.
"It's under investigation. If we did something wrong, it's normal to pay," added the 46-year-old Argentinian.