The Gers attack weren't at their best as St. Johnstone progressed in the League Cup at their expense on Tuesday night.
The Gers went into the Ibrox clash as favourites despite their opponent’s top-flight status, but were quickly brought back down to earth after their storming start to the season.
Two first half goals from a rampant Saints side made the night difficult, before pacey frontman Michael O’Halloran added a third at the restart to give Rangers a nigh-on impossible task.
Tavernier found the net in the 62nd minute, which set the stage for a comeback, only for his attacking team-mates to miss several gilt-edged chances.
“I thought we should have done better for the goals though and we gave ourselves a bit of a mountain to climb conceding when we did in the second half,” Tavernier told Rangers TV.
“But we played well and stuck to our game plan, we just need to brush up on our quality in the final third but we never gave up and we went to the end.
“We created good enough chances but we maybe need a bit more quality in the final third.”
The former Wigan Athletic man has been a revelation for Rangers, and Tuesday’s strike was a seventh of the season from the right-back.
His contribution wasn’t enough to prevent an early cup exit, however, and Mark Warburton will be focused on putting things right for the trip to Morton on Sunday.