In an incredible knockout round play-off encounter at Signal Iduna Park, the Scottish champions raced into a two-goal interval lead through a VAR-awarded penalty from the Gers captain and an Alfredo Morelos tap-in.
Light Blues midfielder John Lundstram curled in a wonderful third after 49 minutes, only for Dortmund’s teenage England midfielder Jude Bellingham to reduce the deficit a minute later with a similar counter.
A Morelos strike was then inadvertently helped into the net by home defender Dan-Axel Zagadou – who had earlier handled for the the penalty – before Raphael Guerreiro’s 82nd-minute effort pulled the German giants closer again.
Despite pulling off an upset in Dortmund, Tavernier warned against making too many assumptions about the second leg at Ibrox next Thursday night.
He said: “We got the result we wanted to take back home, but it is only halfway through.
“There is a lot of positives to take from this game. I thought the boys were exceptional.
“Every man, even the substitutes who came on, everyone did a fantastic job and, like I say, it is only halfway.
“We have a lot of work to do next week, but we put ourselves in a really good position”
On his penalty, eventually awarded after his corner was handled by Zagadou, the Light Blues full-back said: “It was the wait for the VAR. I didn’t see the handball, but when the ref blows you know something is going to happen
“That’s the benefit of having VAR. There was also a bit of a wait before the whistle went (to take the penalty), but I was calm when I took it and happy to give the boys the lead and something to build on.”