After a lacklustre first half, Tottenham were boosted by the dismissal of Jonjo Shelvey early in the second period for a petulant stamp on Dele Alli, who subsequently opened the scoring on the hour mark.
Ben Davies doubled the lead with less than 20 minutes to go, before last season's Golden Boot winner Harry Kane hit the post late on.
"If Harry Kane had scored to make it 3-0 at the end, that would have been a bit more fair," Pochettino told BBC Sport.
"We knew how they play, so narrow and deep. It was difficult to move the ball quickly. We had control and we were helped by the red card but I think we deserved to win the game.
"It was not easy to find the space and create chances. It's important to improve. In a stadium that doesn't have good memories for us, it was important to win. It was a clear red card."
Pochettino also clarified his earlier comments which appeared to suggest debutant right-back Kyle Walker-Peters was not ready, simply explaining that it was unrealistic to expect the 20-year-old to play every game.
"For a young player, don't put pressure on him. He showed in the USA that he was ready to play," Pochettino said.
"It’s important to keep his feet on the grass, it’s only one Premier League game."