The meeting at Stamford Bridge ended 2-1 in favor of the guests.
The victory for Liverpool was brought by goals by Trent Alexander-Arnold and Roberto Firmino, who scored in the 14th and 30th minutes. Chelsea replied with a goal to N'Golo Kante in the 71st minute.
In the seventh round, Liverpool away will play against Sheffield United, while Chelsea will host Brighton and Hove Albion in their field.
Earlier it was reported that Arsenal in the minority snatched out victory in the Premier League match with Aston Villa.