Spurs were unbeaten in the Premier League since going down 4-1 to the champions-elect at the Etihad Stadium in December, but the same opponents put a tough week behind them in style at Wembley.
Gabriel Jesus opened the scoring in the 22nd minute before Ilkay Gundogan converted from the penalty spot three minutes later.
Christian Eriksen reduced the arrears shortly before half-time but Tottenham were unable to capitalise upon a shaky opening to the second period from City, who sealed the points through Raheem Sterling 18 minutes from time.
"We started to play after we conceded the second goal. That disappoints us the most," Pochettino told Sky Sports.
"City were good from the beginning and were more aggressive than us.
"After that we started to control the game, we scored through Christian and then in the second half we started really well.
"That's the way we wanted to play but the first 25 minutes is not the way we try to play or the way we prepared in the week."
Replays showed Hugo Lloris made contact with Sterling fractionally outside the area when he lunged in to concede the penalty but Pochettino refused to seize upon that incident as an excuse.
"I will not complain," he said. "Many things happen in the whole season. It is difficult for the referee to see. That is not a situation to change the game too much.
"At this level, when you play a team like Manchester City, from the beginning to the end you need to compete at your best level. We did not do that in the first 25 minutes."