Right-back Naughton was sent off by referee Chris Foy just before the half-hour mark in the Premier League clash at Upton Park on Saturday.
The Spurs defender raised his arms to block a Kevin Nolan shot on the turn at close range in the penalty area. Foy pointed to the spot before dismissing Naughton, with Mark Noble missing the resulting penalty.
After James Collins was also sent off in the second half, debutant Eric Dier ran through to win the game for Pochettino's side in stoppage time.
The Argentinian coach told Sky Sports: "I think the red card was unfair. It’s possible the club will review the (video) clip and something will happen."
However, Pochettino was pleased with the reaction from his players after Naughton's early exit, adding: "The more important thing is that the team kept calm, carried on and never gave up."
Spurs lost all three meetings with Allardyce's West Ham in league and cup last season, and Pochettino admitted it was a huge boost to take all three points in his first competitive game in charge.
“It’s a big win, and it was a very tough game," he continued.
“We are very happy, and I congratulate my players. The support was unbelievable too. But this is only the first step.”
On goal hero Dier's debut, he said: "I’m happy for him on his debut in the Premier League, but also for the team. We showed the spirit that we want to show in every game.
"We played like a team. This is the important thing. To win three points is important too, but going to 10 versus 11, we showed we believe - and we changed the game in the last minute.
"Even when it was 10 versus 11, we believed it was possible to win. This is our philosophy."
Tottenham travel to Cyprus to face AEL Limassol in a Europa League play-off round first-leg tie in midweek, before returning to Premier League action at home to Harry Redknapp's QPR the following Sunday.