The full-back, who captained the club for the first time in Sunday's 6-1 win over Ayr United in the Scottish Cup, has extended his stay until the summer of 2021.
Tavernier joined Rangers in July 2015 from Wigan Athletic and made an immediate impact, scoring on his debut against Hibernian in the Challenge Cup.
The 26-year-old told the Gers website: "I am on social media and I have had a great response from the fans.
"It feels like everything is going in the right direction at the moment, it felt right on Sunday, it was a perfect moment and this new deal is the icing on the cake.
"I would love to be the captain of this club, it is a great responsibility but I think I have grown into a player who can take the pressure on his shoulders."
The right-back has made 126 appearances for the Scottish Premiership club, scoring 23 goals in that time, and has claimed six goals and six assists this season.