The Light Blues showed their powers of recovery over the course of a frenetic 90 minutes, coming from behind after Kevin Holt dispatched from the penalty spot for Thistle in the first half.
Speaking to RangersTV after the match, Tavernier reflected on an eventful cup tie in Govan: "We’re through to the quarter-finals, but we’re disappointed with the performance. It’s not the performance that we wanted today.
"We were definitely below par and we made it harder for ourselves than we needed to. We'll look back at this game, work on things that we need to improve which we know as a team that we need to improve. The main thing as a team is that we're through to the next round.
"The subs made a big impact on the game. We wasted 45 minutes in the first half. We made it harder for ourselves conceding the penalty and we go in 1-0 down, but the lads came on and made an impact. There was a slight improvement in the second half and we're through to the next round."
There was a bizarre moment midway through the second half when, unaware to Malik Tillman who was on the ground after a strong tackle, he pinched a loose ball from Thistle’s restarting throw-in as they appeared to be gifting possession back to Rangers before racing through to score.
After discussion between the bench and the players, Thistle were allowed to walk the ball into the goal to level the match.
Tavernier commented: "We have high standards. It was the right thing to do. I don't believe Malik knew what was happening. He's an honest lad. He was on the floor at the time, and he didn't know if the ball went out or if we kicked it out. We knew we had enough to win the game and we did."