Romelu Lukaku's strike proved to be the difference at Goodison Park after second-half substitute Gaston Ramirez had cancelled out Seamus Coleman's ninth-minute opener.
The result left the Saints ninth in the Premier League table after just one win in December, but Pochettino believes that there were positives to take from the loss on Merseyside.
"I think overall the game was a very good game with both teams proposing attacking football. It was good going forward football," he said.
"It is a shame because we leave with nothing and I think we deserved a lot more from this game.
"I think overall the team played very well and there were many phases of the game where we were superior to Everton.
"It's clear that we are a young team but it's a team that has a personality, it has a certain attacking style that it tries to always impose that attacking style wherever it's playing.
"It always tries to go for the three points, to go for the win."
The Christmas period now sees Southampton face Chelsea on New Year's Day and Pochettino acknowledged the need to make sure that his players are fit enough as the hectic schedule draws to a close.
"It's very important also that we try to recover the players who are missing and that will make us a lot stronger than we already are," he said.
"We all saw what happened in this game and I think we deserved a lot more from this game.
"From the three shots they had on goal they scored twice.
"We created a lot of chances for ourselves but in the end we came away with nothing."