Pochettino's side laboured throughout the first half at Upton Park and were unimpressive in the Argentine's first Premier League game in charge.
The visitors were up against it in the London derby when Kyle Naughton was dismissed for handball with only 29 minutes gone, but Mark Noble missed the resulting penalty.
West Ham were also reduced to 10 men in the second half as James Collins was shown a second yellow card, then debutant Eric Dier stunned the home supporters by scoring a late winner.
Pochettino felt Naughton's dismissal was harsh, but was delighted to secure all three points right at the death.
"We are very happy, so congratulations to players and fans," he said. "They gave unbelievable effort .The support was unbelievable today as well.
"I think the red card was unfair, but the most important thing is that we never gave up, we stayed calm, we believed in our skill and our team, but we kept our offensive play.
"We reorganised our system and carried on. This is football and I am very happy to get three points today.
"This [kind of harsh decision] sometimes happens. It was unfair because the ball was going away from target. It was unfair, the sending off, but it was a penalty.
"Pochettino was also excited by the performance of match-winner Dier, a recent signing from Sporting Lisbon.
The former Southampton boss is not expecting too much, too soon from the England Under-21 defender despite the impact he made on his debut.
Pochettino added: "He is still young, but he showed in the game last week [friendly against Schalke] and today a great performance.
"He is calm and has quality potential, but he is young. We need to give him opportunities to play, but we must be careful as he is young. But today he was the hero.
"He doesn't play like a centre forward in training. I am very pleased for him. It was his debut and he has, like all of the team, put in a great performance.
"This is our philosophy; to believe. We always have to believe for 90, 95 minutes in a game."
