The Ibrox club head to Livingston on Sunday just three days after defeating Porto to enhance their chances of progressing from the Europa League group phase.
Tavernier is savouring Continental football but said Rangers would not be neglecting their domestic chores.
He said: "First and foremost it's about trying to compete for the league. European football is a great addition for the club and the fans.
"It's a big bonus for us. But everyone is here primarily for the league so we need to get three points at the weekend.
"We don't want to be going into the international break having dropped any points because it definitely lingers when you do. The Porto game will be certainly forgotten about (if they drop points).
"We have to keep the momentum going. We need to get a level of consistency. Last season we had speedbumps along the way that cost us. We can't afford to let that happen (on Sunday)."
Despite that domestic focus, the Rangers captain still felt the win over Porto was one of the highlights of his four-year spell at Ibrox.
He added: "That ranks as one of the best results in my time at the club. Porto are a very difficult side, a Champions League (calibre) side. When we played them away from home in the first game you could see the level of class they had.
"But we knew with the full backing of our fans and with the sort of performance that we delivered in the second half that we could get a result. So we're delighted that we kept a clean sheet and scored two."