Borna Barisic was on target late in the first-half in the 1-0 over St Johnstone at McDiarmid Park as Rangers booked their place in today’s draw.
Speaking to RangersTV at full-time, Tavernier praised the squad for their performance. He said: “I am very pleased, that is the job that we wanted to do today to get through to the next round.
"It wasn't the prettiest of games but we left with a clean sheet and we are finding different ways of winning games and I am really happy with it.
“We didn't really create the chances we wanted to and test the keeper in the first-half, it was good build-up play from Fashion and Malik for the goal and it came off the post and what a great hit from Borna.
“We were really happy to go in at half time and the message was to see the game out, not concede and try and obviously create chances which we did and their keeper pulled off a couple of good saves.
“If you look at the pitches we have had to play on we have managed them really well and we have got the three wins that we needed so it has been a perfect week and now we have got a week to prepare for St Johnstone at home.”
With a clean sheet in Perth, Tavernier said it is pleasing for the whole squad as they managed to limit the chances of the home side over the 90 minutes.
He added: “It is important for the back four but not just for the back four, for the whole team.
“We defend from the front, so it is important for the team that we are all digging in together and putting in a shift to try and give them no chances and we are just happy that we got the clean sheet today.”
Tavernier ended with a word for the fans who travelled in their numbers to support the squad this week.
He continued: “It is unbelievable to see, you don't get many clubs where you can fill the whole stadium at an away ground and that is just our fans.
“They are one of the best in the world and we are really happy to have them here supporting us.”