Tavernier scored Bournemouth’s opening goal as they led 1-0 at half time, getting on the end of a magnificent Milos Kerkez cross.
The Hungarian intercepted a Spurs pass deep in the Cherries half and surged down the left, having time to get his head up and pick out the perfect cross for Tavernier to finish.
Evanilson added a second for the Cherries but Spurs came back through Pape Matar Sarr and Son Heung-Min scored a penalty with six minutes to play after he was brought down in the box by Kepa.
Speaking to Match of the Day after the game, Tavernier said: “Everyone in there is disappointed.
(Image: Bradley Collyer/PA Wire) “We came here with the intention to get three points. We went 2-0 up, but we couldn't see out the game and going to have to reflect on some of the mistakes we made. We played a good game.
“We should have taken our chances more in the first half, but overall we're disappointed with the result today.”
On his goal, Tavernier said: “Milos has always been looking for me.
“I know as soon as he gets his head up, he's going to cross the ball and a great ball and happy just to be at the end of it and put it in the back of the net.”
Bournemouth flew out of the blocks, having two big chances in the first five minutes as their press gave them rewards high up the pitch early on.
(Image: Bradley Collyer/PA Wire) “You can see the intensity we started with,” Tavernier said.
“We knew it was going to be one of those games where we both play very high intensity and we played really well to start off with, but we couldn't take them early chances.
“They got their confidence into the game, but we went two up and couldn't see out the game ultimately.
“We have to reflect on that and come back.”
The result sees the Cherries still in the pack of teams in contention for the European places with ten games to play, with five points separating fifth from tenth.
(Image: Bradley Collyer/PA Wire) Asked if leaving Spurs with only a point is a sign of how far the Cherries have come this season, Tavernier said: “Definitely.
“We know that we're having a great season at the moment, but ultimately we're disappointed with the performance today and how it ended. We know what we're playing for. We're playing for big things.
“So we’ve got to push to the end of season, but we can't dwell on it too much and we have to focus on next week now.”
On if they are confident on completing European qualification next season, Tavernier said: “We have to take it a game at a time.
“We know what we need to do and how we need to do it and we just have to reflect on today and correct those wrongdoings.”