At 17 years and 45 days, Woodburn became Liverpool's youngest ever goalscorer when he netted in the 2-0 win over Leeds United in the EFL Cup quarter-finals on Tuesday night.
Manager Jurgen Klopp is likely to keep Woodburn on the fringes of the first-team squad but Mane says that the midfielder has the potential to have a "very good future" with the Merseyside giants.
The 24-year-old told PA: "I want to congratulate [Woodburn]. He is a very good player and has a very good future.
"I hope he will keep going and working hard and I think he will be a great player. Everyone was happy for him to score his first goal, especially in a big atmosphere at Anfield."
Before his appearance on Tuesday, Woodburn had been handed his Premier League debut during the closing stages of the 2-0 triumph over Sunderland at the weekend.