The Serbian has had to wait a long time to make his mark after signing for Jurgen Klopp's side back in January.
Grujic said: "It was an incredible night for me, making my first competitive start and a first win.
"It was a good game, a 3-0 against Derby, they are a good team.
"We played well and I want to say thank you to all the fans who travelled away, it was a nice atmosphere.
"I had a good pre-season and then I have been on the bench but I'm still learning and adapting to English football.
"Every day, I try to be better and better and I think my debut was a solid game.
"I will try to continue learning and working hard and I'm sure that I will get a chance again. [I try] to make myself physically [better] and every day I work with the staff.
"I'm still a young guy, one of the youngest in the squad, and I will be better and better."