The Northern Irishman put in an influential performance as Kilmarnock were defeated 2-0 with second-half goals from Antonio Colak and Alfredo Morelos.
Davis could have left in the summer when his contract was set to expire but penned a one-year contract extension.
The win at Ibrox against a stubborn Killie side was the 37-year-old's first appearance of the season.
“We had some players who played really well and gave the team what was needed," Van Bronckhorst, who hopes to have Ryan Kent back for Tuesday's Champions League qualifier, said. “Steven Davis it was his first start this season but you still see the ability and quality that he has. He showed that again today. I’m pleased with his performance.
"Of course he was one of the players after last season that we had to sit down with him and talk about his future. With the age he has, it is always good to leave it until the end of the season and he felt he had more in him for his future
"He played really well and he is a player who is a great example for all my players in the squad because of the way he is being a pro and making sure he is fit. You could see, if you do so then you can play at a really high level at a high age.
"I am really happy he stayed on because you have to be mentally ready to play a new season and a year. He looked strong and he is just being himself. The performance was really good.”
Playing against a deep block
Ridvan Yilmaz was handed his debut at left-back before being replaced by Borna Barisic in the second half. Van Bronckhorst felt the Turkish international “maybe needs a little bit more time to adjust to our playing style”.
He expressed his happiness with the performances of Tom Lawarence and Malik Tillman. The former's strength of playing “really well in the pockets” was noted, as was the latter’s German’s “technical ability”.
Tillman, especially, impressed with his first-half performance, getting on the ball and creating space for himself.
“Those players you need in the team to create if you’re playing against a deep block," Van Bronckhorst said.
“He came in a couple of weeks ago and has been welcomed by the team really well. You can see his talents are really high from the academy of Bayern Munich and also played competitive games for them. You can see the quality that he has.
"He’s a strong player, you could see that today. Some players when they come to play in this league struggle a little bit in the beginning but he’s not had any problems with that. And his technical ability is of a very good standard.
"Today he created some chances for us. Just before he came off there was his pass to Morelos. It’s a player who is capable of creating those chances for us.”