Philippe Coutinho's hat-trick, along with a brace from Sadio Mane and a goal each from Roberto Firmino and Mohamed Salah, meant Liverpool finished top of Group E and qualified for the knockout stages of the Champions League for the first time since 2008/09.
The massive win at Anfield also meant that Liverpool are the highest-scoring English side in the history of the Champions League group stages with 23 goals, two more than Keane's United managed in 1999.
Keane advised Liverpool to keep their feet on the ground and warned that tougher tests await in Europe.
"Their attacking players enjoyed it but the big tests are yet to come for Liverpool, clearly they've beaten these so called lesser teams, but they've beaten nobody yet," the former Red Devils captain was quoted as saying by the Liverpool Echo.
"It's lovely to watch but they wouldn't get that time and space against a decent team, it's as simple as that.
"Give quality players time on the ball like that and they are going to punish you. The reality is if they are playing a top team in Europe they won't get the time and space, they won't get those opportunities.
"If you know this game then you understand they have so much bigger and tougher tests coming up."
Former Arsenal defender Lee Dixon said Liverpool's back four remained an area of weakness that stronger opponents would be able to exploit.
"It wasn't that long ago we were talking about Liverpool's defence and how many goals they've let in, how the back four were all over the place," Dixon said.
"I think they have improved in that but I don't necessarily think they've been tested enough to know that's gone away."