Tottenham left the Santiago Bernabeu with a point after holding two-time defending champions Madrid to a 1-1 draw on Tuesday.
It was an enthralling encounter between the top two teams in Group H after Raphael Varane's own goal was cancelled out by Cristiano Ronaldo in the first half.
Madrid's pursuit of a second goal was thwarted by Tottenham captain and goalkeeper Hugo Lloris, while Harry Kane and Spurs were denied by the gloves of Keylor Navas.
Tottenham manager Pochettino had nothing but praise for his players post-match in the Spanish capital, saying: "I'm really proud and so happy. Everybody wanted to compete and show ourselves that we can compete at this level. I am so happy.
"The performance was fantastic. I congratulate everyone because it was very good and of course our fans were fantastic, unbelievable and [I say] thank you so much for the support and the help. It was unbelievable."
Pochettino added: "Always I tell the players it's a game to enjoy – play, run, feel free, try to play because [when] you play in that level it's our dream, always, from when we started three years ago, to play in the Champions League, against Real Madrid and in the Bernabeu. Football is about enjoying.
"Being brave is so important. If you want to win and if you want to increase your level, learn and improve, it's so important to be brave.
"Of course, when you play you must keep your identity and compete. In the Champions League last season that disappointed us that we didn't compete but in these three games [against Borussia Dortmund, APOEL and Real Madrid] I think the team is more mature and we don’t just play, we compete and that is most important."
"[We are in] a very good position," he continued. "We are on seven points with Real Madrid. It's fantastic. Our challenge is to reach the next stage and we are happy with the result. I'm very happy for our fans, for our team and the squad. I'm so, so happy."