The skipper played his part in the 1-0 win as his pressure from Ianis Hagi’s cross, saw Saints midfielder Jason Holt turn the ball into his own net.
It makes it four wins and two draws for the Light Blues in the last six outings in all competitions, including Thursday’s 4-1 hammering of the French side.
Tavernier was delighted to secure all three points but believes the squad have to be more clinical in front of goal – starting with Kilmarnock at Ibrox on Wednesday.
Speaking to RangersTV, he said: “We got three points, a clean sheet and we had better management towards the end of the game.
“Jack Butland has had one save to make and that is good from the team’s point of view.
“It was important off the back of a good European win to keep that momentum going and get the result we wanted and we obviously did that.
“It is different types of games from Europe compared to domestic matches. St Johnstone played some good football and asked different questions of us, but we manged it well.
“We had to be patient and it was important we did that. There were chances there that we have to got to finish.
“We have to be more clinical and we need to improve on that, like we were on Thursday against Nice.
“It was important to come here and get the three points before looking forward to Kilmarnock on Wednesday.
“We didn’t see out the game well enough against Hearts and it is all about game management.
“You want more goals but sometimes you have to shut up shop and control the game, limiting the chances they have.
“We did that better and it is something we want to keep improving on.
“We have to continue to work hard on what we have to do and earn the right to play and win the games.
“When the first whistle goes on Wednesday against Kilmarnock we have to earn the right to win the game and that’s what we will be looking to do.”
The captain returned to the starting line-up after being among the substitutes for the matches with Dundee United and Nice.
Being rotated from time to time is a new experience but Tavernier insists no matter who starts matches, he will be demanding the highest standards alongside boss Philippe Clement.
He added: “It is different being rotated and it is something that is new to me.
“I have to stay professional and push myself and the team so we can be better and better every day.
“Anyone who is asked to start in the team has to give their all and that’s what the manager requires and what I require as the captain.
“The lads are performing but we have to keep pushing ourselves.”