The North Londoners took the lead after an hour of Sunday's Premier League encounter through Dele Alli, not long after the Magpies saw Jonjo Shelvey sent off for a stamp on the Spurs midfielder.
Ben Davies then added a second on 70 minutes, and in injury time, Harry Kane's low strike inside the area from Son Heung-min's pass cannoned off the foot of the Newcastle post.
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 [for Shelvey].
"If Harry Kane had scored to make it 3-0 at the end, that would have been a bit more fair."
Spurs have now won 13 of their past 14 Premier League games, winning each of their past four and scoring 17 goals in the process.