Cyriel Dessers set the Ibrox men on their way to victory in the first half with his sixth goal of the season before captain Tavernier sealed the points with a second-half spot-kick after he had sent his earlier attempt from 12 yards out wide of the post.
Clement has now overseen seven games unbeaten – six of which have brought victories – since replacing Michael Beale last month.
After four consecutive defeats left his side bottom of the table prior to the visit of the on-form Ibrox men, Livi boss David Martindale made five changes to the team that lost 1-0 to Dundee last Sunday.
Goalkeeper Jack Hamilton, Miles Welch-Hayes, Jason Holt, Danny Lloyd and Stephen Kelly all dropped out to make way for Shamal George, Cristian Montano, Mo Sangare, Bruce Anderson and Luiyi de Lucas.
Clement freshened up his Rangers team for the trip to West Lothian, making four alterations to the XI that started Thursday night’s 2-1 win at home to Sparta Prague in the Europa League.
Ben Davies, Todd Cantwell, Sam Lammers and Danilo dropped out to be replaced by Leon Balogun, Dessers, Tom Lawrence and McCausland.
After enjoying a bright opening, the Ibrox side thought they had gone ahead in the 16th minute when Connor Goldson had the ball in the net from close range.
However, the goal was ruled out following a VAR check after Dessers was deemed to have been offside and interfering with play as he darted in at the near post and tried to get a touch on Tavernier’s inswinging free-kick. The wrong-footed Livi keeper George then pushed the ball out to Balogun, who in turn nodded it into the path of fellow centre-back Goldson.
Rangers created a good opening with a rapid counter-attack three minutes later but McCausland shot over after bursting on to a Lawrence pass.
The 20-year-old winger’s pace caused more problems for the Lions midway through the first half when he won a penalty after being brought down by George as he ran on to a Tavernier pass.
Tavernier, however, sent his spot kick wide of the goalkeeper’s left-hand post.
Rangers had been well on top and they made a deserved breakthrough in the 26th minute when a lovely pass from Lawrence released Nigerian forward Dessers, who got away from Sean Kelly to slot the ball behind the advancing George.
McCausland thought he had scored his first senior goal for Rangers when he fired the ball high into the net just before half-time but the goal was disallowed for a foul by Abdallah Sima.
Rangers continued to dominate after the break but – after struggling to create many clear chances – they had to wait until the 74th minute to put the game beyond their hosts.
Following a VAR check, it was deemed that the ball struck the arm of Michael Devlin as the Livi captain challenged Rangers forwards Danilo and Sima in the air while trying to defend a cross from the left from Borna Barisic.
This time Tavernier made no mistake in tucking the ball to the right of the helpless George as Rangers continued their strong start to Clement’s reign and left Livi at the foot of the table on goal difference from Ross County.