The academy graduate, who at the age of 17 years and 45 days beat Michael Owen’s record by 98 days, was just 14 minutes into his senior career when he blasted home on front of the Kop.
“I want to congratulate him. He is a very good player and has a very good future,” Mane told Press Association Sport.
“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.”
Woodburn, who made his first-team debut with a cameo lasting barely a minute against Sunderland on Saturday, came off the bench to have an impact at Anfield on Wednesday night.
However, in a team of eight changes it was right-back Trent Alexander-Arnold, an 18-year-old academy contemporary of Woodburn, and 19-year-old Ovie Ejaria, making his fourth first-team appearance, who were the stand-out players in the starting line-up.
Mane believes the youngsters have added an extra dimension to the dressing room.
“We have a great team this season and all the young players are playing confidently and are doing well and the game will become easier for them,” he added.
“I think they will help the team to move up.”
Klopp has put his faith in a number of youngsters during his 14 months in charge and has been rewarded with several impressive performances, but he is keen to play down the growing expectation on them.
“We don’t even talk any more about Ovie because he’s 19!” he said.
“All the young boys did very well – 17, 18, 19, they did well in a very important game.”
Woodburn’s emergence as an option to provide occasional cover up front is timely after the Reds lost Philippe Coutinho for at least five weeks with ankle ligament damage, with Daniel Sturridge currently sidelined by a calf problem.
Striker Danny Ings has already been ruled out for the remainder of the season, but at least Klopp knows he has backup to call on should he need it, as he looks to keep Liverpool’s title challenge on track during a busy December, with a two-legged EFL Cup semi-final and FA Cup involvement to come in January.
“It’s a good moment for Liverpool, but it’s difficult too because there’s a lot of work to do and a lot of games to play,” he added.
“Four weeks ago we had no clean sheets. Now we start getting clean sheets, but we don’t any more score six goals in a game and all that stuff.
“We have problems, we have injuries, we have strong opponents, but we have wonderful people around us, we have a wonderful crowd, a wonderful stadium, wonderful away supporters.
“We go now to Bournemouth – a small stadium, but very intense with a good team.
“We are in a good moment until now, but we have to carry on and that’s how it is. Hopefully it stays like this.”