The Premier League outfit won 1-0 against Rovers in their first pre-season friendly match.
England international forward Danny Ings scored the only goal of the game at Prenton Park in the 79th minute.
Summer signing Mane featured for Liverpool in the match, and his performance pleased club manager Klopp.
“I was quite satisfied,” Klopp told Liverpool’s official website about Mane’s performance against Rovers. “He had an exciting week or 10 days.
“He trained a lot in his break, did what he had to do and now he is here. He is a very, very nice guy and a good football player.
“Everybody could see in the first half, with really tired legs he really played a good game. He could have scored, could have done one or the other thing a little bit better, but he was fine.”
Mane recently joined Merseyside outfit Liverpool from Premier League rivals Southampton in the summer transfer window.
According to BBC Sport, the Reds had to pay the Saints a transfer fee of £34 million to secure the services of the Senegal international winger. It was reported that the transfer fee could rise to £36m.
The 24-year-old winger scored 11 goals and created 41 chances in 37 Premier League matches for Southampton during the 2015-16 season.
The former Red Bull Salzburg winger had a pass accuracy of 81% and a shot accuracy of 52%, and won 26 headers, 77 dribbles and 39 tackles.