Playmaker Christian Eriksen struck a late winner after teenage substitute Kelechi Iheanacho had levelled to cancel out Harry Kane’s opener from the penalty spot following referee Mark Clattenburg’s decision to penalise Raheem Sterling for handball.
Victory saw Spurs regain second place in the league after it had been temporarily claimed by north London rivals Arsenal and move to within two points of leaders Leicester.
“The supporters are right to dream, it is one of the keys to the club,” said Pochettino. “They show big support from the beginning of the season and I say thank you to them. It is important for our supporters to believe.
“I think that it was a game that was very important. You could feel it was not a normal game and a real test for us, for that we feel very pleased for the players.
“The mentality was good and a perfect performance against a team with a lot of quality players who are top class. Full credit to my players.”
City have now lost successive matches at the Etihad Stadium to fellow title contenders and sit fourth after their latest setback, six points adrift of top spot, although boss Manuel Pellegrini insisted the fight for supremacy would continue.
“We needed to get the three points," said Pellegrini. "We are not going to give up but Leicester and Tottenham are fighting for the title. We need to keep working. The spirit of the team is very good."
“We are disappointed because we played better than Tottenham. They did not have many chances to score until we started to take risks. Before that, it was a close game and the decisive penalty was important.”
After a chanceless and sedate first half, Tottenham opened the scoring eight minutes into the second period as Sterling was adjudged to have handled despite turning his back as he jumped to block a Danny Rose cross.
Kane fired his spot-kick down the centre of Joe Hart’s goal for his 16th league strike of the campaign before Yaya Toure’s free-kick rattled the Tottenham crossbar moments later.
Pellegrini threw on Iheanacho and the 19-year-old forward repaid his manager’s faith by equalising with 16 minutes remaining as he confidently swept home Gael Clichy’s centre from 12 yards.
But the final word belonged to Eriksen on his 24th birthday. The Denmark midfielder collected Erik Lamela’s pass, which nutmegged City centre-half Nicolas Otamendi, before firing past a helpless Hart.