The 22-year-old joined the Reds from Swansea City on Friday following an impressive 2011-12 Premier League season with the Swans.
Molby, who featured in 292 games for Liverpool in the 1980s and 1990s, is certain that manager Brendan Rodgers has signed the right kind of player to impose his passing football style at Merseyside.
"If you want to be a good passing team, you need to have someone to start it all," the Dane told Liverpool's official website.
"A few years ago, Liverpool had Xabi Alonso, who used to start our moves for us. Joe Allen is that type of player; he gets on the ball, he gets things moving, he's very, very sharp and he's got two good feet.
"But he can also cope with the physical demands of the modern game and, as part of a three-man midfield, he can get up and down the pitch. I think he's going to be a very good player but I think first and foremost, he's been signed because he's going to be the catalyst to get our play going."
Molby, 49, is confident that Allen will establish himself as one of the greatest players in Liverpool's history.
"He can definitely rise to the challenge otherwise Rodgers would not have signed him," he added. "No disrespect to Swansea, but Allen is now playing at a bigger club with bigger players and Brendan feels he can step up.
"I think he's got the ability to become a top-class player in the years to come. I think that's why we signed him."
Liverpool will begin their Premier League campaign with an away game against West Bromwich Albion on Saturday, August 18.