The champions claimed a 1-0 win at Wembley on Monday thanks to a sixth-minute strike from Mahrez, who dedicated his goal to former Leicester City chairman Vichai Srivaddhanaprabha after he died in a helicopter crash.
Spurs competed well on a pitch left in ruinous condition by Sunday's NFL match, with the surface seemingly taking its toll when Harry Kane and Erik Lamela both wasted clear chances to equalise with poor touches.
Pochettino refused to blame the turf but did admit he was annoyed to concede so early in the match after Kieran Trippier misjudged a simple long ball from goalkeeper Ederson.
"The circumstances of the pitch was the same for both sides," he told a news conference. "I think it was a very competitive game. We conceded a goal so early that was so cheap, and football punishes you.
"I am so proud, though. The effort was fantastic. Last season, [it] was difficult to compete with City, the gap was massive. Today, it was competitive.
"Yes, [they were] a little bit lucky because when they made a mistake we didn't score, and when we did, they scored in one action. I'm disappointed in one way with the result, but happy with the performance."
Speaking further of the difficult conditions, Pochettino said: "I don't have the feeling that we could not play. Both teams tried to play in the way we normally play.
"It's difficult to dominate them, because they play a similar philosophy to us, trying to dominate the opponent. Of course, I think [in the] first half we suffered a little bit because to concede so early it's normal that they play with more security.
"But I think in the second half we dominated, it was very good. I am happy with the performance."
However, Pochettino does not think his side should be judged in the same bracket as the reigning champions.
"We cannot compare ourselves to Man City," he said. "They are on a different level. They have won the Premier League title. We are focused on trying to win games."