Sanchez was shown a straight red for elbowing Richarlison in the 52nd minute, damaging Spurs' chances of claiming a first win in four Premier League matches.
Pochettino, who refused to be drawn on whether he felt Sanchez's dismissal was justified, applauded his players' effort.
"I am so pleased. I am very frustrated with the results recently but I have nothing [negative] to say about the performance," the Argentine said at his post-match press conference.
"[The players] were always thinking of going forward and scoring to win the game. We did not concede one chance in the second half with 10 men and I think we were close to scoring.
"I am so proud of their character."
1 - Spurs have kept just one clean sheet in their last seven Premier League games, after registering four shut-outs in the five games before that. Leak. — OptaJoe (@OptaJoe) December 2, 2017
Pochettino also downplayed Dele Alli's apparent unhappiness over being substituted in the 86th minute.
The midfielder was seen shaking his head as he followed Christian Eriksen and goal-scorer Son Heung-min to the bench.
"Like Christian or Sonny, the players who have character always show their frustration," he said.
"Today was a game we wanted to win. Because [Alli] had to play deeper with Mousa Dembele, the effort was amazing from him and the whole team."