It was a gripping spectacle right until the final few minutes when City had a stoppage-time goal from Raheem Sterling, which would have sent them through, ruled out for offside after a VAR review.
"We are all tired after an amazing, crazy and unbelievable night," Pochettino told a news conference on Thursday.
"It means a lot (to reach the semi-finals). We still haven't realised because we are still in a dream. We're still in a bubble. We can feel the happiness of our fans, also through the messages of family and friends.
"But because we are so focused and busy preparing for Saturday, you do not allow yourself to live your reality. Our job is to be focused and concentrated and use all the tools to be ready for Saturday."
Spurs, currently third in the league, return to the Etihad Stadium on Saturday as they hope to boost their chances of finishing in the top-four for a fourth consecutive season.
"In 10 days to play three games... everybody thinks we know each other better now but there is not much time to train or prepare," Pochettino added.
"We need to assess some players, we need to assess some players that cannot be possible to play.
"That is the most important ... to try to take the best decision and to provide the team with energy to be possible to fight and challenge for the three points."
In their first European Cup semi-final appearance since 1961-62, Tottenham will face Ajax Amsterdam, with the first leg scheduled in London at their new 62,000-capacity stadium on April 30.
"We need to be humble. No one expected us to beat City. It can give massive belief but if you are not clever enough to manage it, that victory can make you weaker," said Pochettino.
"It's so important to respect Ajax. Beating Real Madrid and Juventus means they are more favourites than us -- that is a reality."
The Champions League final takes place on June 1 in Madrid and Pochettino has not ruled out the chance of striker Harry Kane recovering from an ankle injury to play in it, should they beat Ajax.
"We are going to assess him day by day. We are not optimistic but you know Harry Kane - with Harry Kane, all is possible," he added.
(Reporting by Hardik Vyas in Bengaluru; Editing by Toby Davis)
