Harry Kane gave Tottenham a superb start with the opening goal in the second minute at the King Power Stadium, but they failed to take control of the game and allowed Leicester back in.
Although Spurs remained ahead at the break, Leonardo Ulloa quickly levelled things up in the second half and the hosts looked more likely to score again.
They failed to do so, though, and a quick piece of thinking from Christian Eriksen caught Ben Hamer out, the Denmark international firing home a long-range free-kick after 70 minutes.
Pochettino admitted Tottenham had ridden their luck at times in the match, so was delighted to have come away with all three points.
"I'm happy for the result," he told Sky Sports. "We started very well, scored early and had many chances to score.
"After 20 minutes we lost control of the game. It was an open game and Leicester played very well.
"Never in the Premier League is it easy. Leicester played very well. You can never relax in the Premier League.
"The level of our league is high and Leicester deserved more, because it didn't reflect their performance on the day."
The win moves Tottenham to within two points of fourth place ahead of their home clash with Manchester United on Sunday.
