The captain scored the all-important goal in Geneva this evening as a 1-1 draw was enough to see Gers into the next round of the competition.
Speaking to RangersTV at full-time, Tavernier said: “I am obviously delighted that we are through to the next round. That was the main objective to get to the next round no matter what cost.
“Obviously we are slightly disappointed that we conceded tonight and with our first-half performance, but I felt that in the second-half we were much better and we are through to the next round - that is the main thing.”
Servette took the lead midway through the first-half in Geneva this to level the aggregate score, but Tavernier was pleased the squad were able to show a reaction after the break.
He continued: “I felt that in the first-half we built through their team really well and got into their final third. We were maybe too quick on our crosses or our passes and we weren't clinical enough with our final pass but we knew we were going to create chances.
“It was just about making sure our defensive shape was right and if one player is slightly out of position then they can obviously play through us but that got fixed in the second half and we were much stronger.
“I thought Jack [Butland] made two incredible saves and kept us in the tie as well but I am just delighted that we are through, and it is something to build on.
“It is my little trademark going into the back post and as soon as it goes to Borna [Barisic] I know he is going to cross it with such quality and I was in the right place at the right time. I am delighted it went into the back of the net.”
With a repeat of last season’s UEFA Champions League play-off, Rangers will once again meet PSV Eindhoven in a bid for a place in the group stage of the competition.
Tavernier says the squad are focused on continuing their journey in the Champions League, he added: “We want Champions League football, that's what we want and we will do everything in our power to try and make sure that happens.
“It is going to obviously be a tough game against PSV, but we are going to take full advantage of our home crowd next Tuesday and make sure we are in a really good position going into the second-leg.”