It was a big performance from Giovanni van Bronckhorst’s side as a 3-1 victory at Ibrox saw them progress to the last four with a 3-2 aggregate win.
Tavernier was on the scoresheet twice in what he described as a ‘special night’ for the club and the fans.
Speaking to RangersTV at full-time, he said: “It was really special, I said yesterday we can make the fans proud and make the club proud and we did that tonight. I thought everyone put in a great shift, we didn't want it to go to extra time but it did and we really dug in.
“There were a few nervy moments but we outrode them and we are in the semi-finals so it is a special night for the club and the fans and you could see when we were going round the pitch at the end of the game just how much it means to everyone.”
Tavernier opened the scoring inside two minutes as Rangers got off to a flying start at Ibrox, but the captain praised the performance of Kemar Roofe who netted the crucial winning goal in extra time.
He continued: “Two minutes in and I found myself at the back post and funny enough I was watching the Leicester game earlier and there was a chance for one of the PSV players and I thought if he slid in he might’ve scored.
“It was in my head just to slide and try and get it on target and obviously that was a great start for us. We were constantly a threat and I thought Kemar was unbelievable tonight.
"He had two offside goals but he got his reward and he was the one who won the penalty as well so it was a great shift from him and I am really proud of him.”
Next up for the Light Blues is an Old Firm Scottish Cup Semi-Final on Sunday and Tavernier says the squad must show the same levels at Hampden.
He added: “You have to dig deep going into extra time, they had 10 men and then obviously went to nine men so it helped us but everyone dug deep.
“We have to rest up and focus on the weekend now because it is a massive game ahead of us and we have got to put that same effort in this weekend.”