The skipper, who made it an incredible seven goals from seven games tonight with the goal which proved to be the winner, couldn’t keep the smile from his face when he spoke afterwards to RangersTV.
He explained: "I am just delighted for the club and the fans and for the application the lads have put in. We have always wanted to get back into the group stages - it was a great test tonight and a big scalp - Galatasaray are a massive club and I am just delighted that we have managed to do it.
"It means so much to us as a team, to the fans and the club that we are back in the group stages so I am over the moon.
"I am loving playing very minute. The gaffer gives us a platform and pushes us on as do the whole management team.
"And, I am playing with a great group of boys that help my game so much. I am just loving football at the minute - I am really taking it in my stride and I just hope I can keep on improving and keep on helping others around us and help the team, that is my main focus, to get us to where we need to be."
With the high standards Gers have, there was natural disappointment with the goal they conceded late on, but barring that, this was a mature performance from the Light Blues who showed how streetwise they have become in European competition.
Tavernier continued: "I was devastated for Greegsy [Allan McGregor] to concede that goal as he's not had anything to do tonight.
"The manner we conceded in was disappointing, but, it is one of those - we are through and we will put it to one side.
"But the discipline in the team is like that on European nights - we are so well drilled in defending and we create as a team - we just need to take that into domestic games now, and if we can, we have a great chance.
"I'm just so proud of everyone in that changing room."
On his goal, which came after Gers moved the ball from one end of the park to the other in a stunning move, he said: "I think that's the second one Borna [Barisic] has put on a plate for me! He has a fantastic delivery and he keeps on putting the balls in for the lads and I am just happy I contributed to helping the team, so I am over the moon."
The only thing really which could have made the evening better would have been to have played the match in front of a full house.
Tavernier was satisfied to have still delighted the supporters in some manner, however, and added: "Scotty [Arfield] was just telling Cedric [Itten] 'if this was a full house, this would have been the best atmosphere you will have heard.'
"These are the nights where you want to be playing in front of 50,000, but I am sure the fans at home were screaming at their TVs all happy we have got through - it is for them at the end of the day.
"Now, we have to recover and focus for Sunday."