Pedro, in his first start for the Blues, scored both goals for the 2-0 win.
Bale, on DAZN, said afterwards: "It's not just the goals, it's the link-up play, the passing and the vision.
"Even just coming into a new team and trying to understand your teammates is very difficult as well. It was a great all round performance and he thoroughly deserved the player of the match.
"I'm not saying Brighton are a small club but he has made the next step up to a team that are playing Champions League and challenging for trophies.
"He is on the right trajectory and I think if he keeps his feet on the ground and keeps working hard - you can see the ability he has so hopefully he can keep ticking on."
Pedro will be full of confidence
Meanwhile, departing Newcastle striker Callum Wilson, also acting as a pundit at the tournament, said: "He's already a player with confidence but he will have wanted to hit the ground running straight away and that's exactly what he has done.
"Now he will have even more confidence because he's scored not one but two goals.
"I preferred the first goal because he's dropped outside the area and has no other thought in his mind other than taking it on. It was a fantastic finish and what a signing he could turn out to be."