A stunning second-half strike from Gaston Ramirez cancelled out Seamus Coleman's early opener, before Romelu Lukaku popped up to wrap up the points for Everton 16 minutes from time.
But Pochettino feels his team should have picked up at least a point at Goodison Park.
"I think overall the game was a very good game, with both teams very good going forward, but it is still a shame because we leave with nothing and deserved more from this game," said the Argentinian.
"I think the team put in a great performance, not just one player, but the team as a whole and I'm very happy with the performance but obviously very disappointed with the result.
"From the three shots they had on goal they scored twice and we created a lot of chances for ourselves but in the end we come away with nothing."
Pochettino also praised goalscorer Ramirez, hinting that the Uruguayan could have a more important role to play in his future plans after finding first-team opportunities hard to come by this season.
"I am very happy with his performance today because he's in the process of recovering his confidence and his attacking style and he's on the right path," he added.
"I think it's clear that he is in a process of adaptation - it's taking him some time to adapt.
"Today he got some minutes and scored a great goal so we hope that with time and with more games he will be able to recover that confidence he needs to be a better player in the future."
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