He also scored the winning goal in Madrid's Copa del Rey final victory over Barcelona in April, and Redknapp has been impressed by how he has adapted to life in Spain.
"Cristiano Ronaldo and Lionel Messi are the two best players in world football, but Gareth Bale is the heir to the throne," Redknapp wrote in his column for the Daily Mail. "There is no doubt he can be the best player on the planet in five years' time."
"What Bale has achieved in his first season at Real Madrid is nothing short of sensational. To go to a new club, a new country and an entirely new culture and perform as he has done is remarkable.
"He was not at his best on Saturday night but big players change games and that's exactly what he's done. He scored a wonder goal to win the Copa del Rey last month and made the crucial breakthrough in extra time in the Champions League."
Bale has scored 22 goals in all competitions for Madrid this season.