Tavernier scored for the Cherries, with goalkeeper Neto outstanding late on.
"We have to be careful what we say but there is a decision that I thought was a goal and we have to stand by it and I think we can be pleased with the performance today," said Tavernier.
"I feel like the first half we were definitely on top. When they got the goal we responded well and created that chance to get the goal and three points but it wasn't meant to be."
"I want to try and score as many goals as I can. Now Dom (Solanke) has left, it is up to us to score more goals and that is what we are trying to do."
On new signing Evanilson, he said: "I think he is technically very good and when the ball is around the box, his finishing is very good. We are expecting good things from him."
On Neto's influence, Tavernier added: "He is the reason we come away with a point today and a big round of applause to him for keeping the ball out of the back of the net."