The Reds, playing at the home of the Boston Red Sox who like Liverpool are owned by the Fenway Sports Group, gave a start to young forward Raheem Sterling.
Goalkeeper Peter Gulacsi twice denied Roma’s Pablo Osvaldo in the opening 20 minutes.
With just over half an hour gone Alberto Aquilani set up Joe Cole but his lob landed on the top of the bar.
United States international Michael Bradley drove post half-time substitute Brad Jones from a tight angle to open the scoring after 63 minutes.
Alessandro Florenzi doubled Roma’s lead with 20 minutes remaining.
Adam drove low into the bottom corner of the goal from the edge of the area with 10 minutes to go but Liverpool were unable to find an equaliser.