The home side took the lead early in the match before Tavernier converted from the spot in the 94th minute to secure a point.
Speaking to RangersTV at full-time, he said: “We are obviously disappointed in the changing room, it is three points we believed we could've got but in the first 20 minutes we weren't close enough with our duels.
“We gave them a couple of chances and they converted one, so we are really disappointed in how we started but after that we responded and tried to create the chances.
"Their keeper made some big saves and we obviously hit the crossbar a couple of times, so we are disappointed we got a point, and we just have to focus on Thursday now.
“That's what Aberdeen are looking for, trying to get a goal and hold on to that, we showed the mentality to continue to keep pushing. Jack didn't have to make a save second-half and we just kept on asking questions and we have just got to be a bit more clinical when we do get those chances in the final third.”
On stepping-up to take the late penalty, Tavernier said: “I was just thinking, just put it in the back of the net, that's what I am asked to do to step up and put it in the back of the net. I felt even when we scored that goal, we were going to get another goal and Sam's chance with his header towards the end I felt we were going to get that winner but it didn't happen, so we just have to learn from this and move forward.”
Looking ahead to the next few weeks, Tavernier insists focus now turns to Thursday’s match against Aris Limassol.
He continued: “It is huge, we have got a lot of games to play until January and it is a massive game at Ibrox on Thursday where we need to get three points to put us in a good position in the group and that is all our main focus now.
"We have to rest and recover and the boys who didn't play today are pushing the lads that are in the team so that is good so the whole team is really at it and we are really pushing each other to get the month that we want.”