The Argentine believes that based on their first half performance, Spurs deserved to win the game at Goodison Park after conceding to former Spurs winger Aaron Lennon in the 22nd minute.
Dele Alli replied for the visitors on the stroke of half-time but they could not find a winning goal which produced a sense of failure in the North London camp.
"We can be proud, it was a fantastic performance," Pochettino told Sky Sports.
"The first half was amazing, an amazing performance, the team played very well.
"We started the second half in the same way, but maybe in the last 15 minutes Everton drew level.
"At the moment, the changing room is down, though, perhaps a little sad because our expectation was to try to win the game. We played a very good game but it was impossible to win.
"The chances we created and the performance I think we deserved more. But this is football, so we need to keep working hard because we are in a good way."