Tavernier’s sixth successful penalty this season took him to 10 goals already this term.
Rangers had only won once in their previous seven league games at Kilmarnock and had lost on four of their past five visits on Premiership duty.
But Steven Gerrard’s side were better this time and Tavernier wrapped up the points in the 19th minute.
Rangers kept their 15th clean sheet of the season despite a spirited second-half showing from the hosts.
The result increased the pressure on champions Celtic, who now have two games in hand.
Gerrard made five changes, with Alfredo Morelos in for his first start in three matches after equalling Ally McCoist’s European goalscoring record with the winner off the bench against Lech Poznan, his 21st goal for the club in the Europa League.
Rangers took control from the start, with Joe Aribo firing just over from 20 yards.
They opened the scoring inside the first 20 minutes for the seventh time in eight league games after referee Andrew Dallas was left with no option but to point to the spot.
Connor Goldson headed on Borna Barisic’s free-kick and the ball hit Ross Millen’s raised hand.
Tavernier put his well-struck penalty into the corner of the net.
Goldson headed Tavernier’s inswinging corner across the face of goal and Aribo saw his shot deflected just wide after Ryan Kent’s knockdown.
Millen denied Morelos with a last-ditch challenge, but Rangers had done enough to maintain their fine form.
