Rodgers is reunited with a player who he took on loan from Chelsea to previous club Swansea City in the second half of the 2010/11 season, which ended in the Welsh club winning promotion to the Premier League via the play-offs.
Borini scored 10 goals in 26 appearances for Roma last season, and he was an unused member of the Italy squad which reached the final of Euro 2012.
It feels really good [to be a Liverpool player]," Borini told the club's official website, "and to be back in England as well.
"It's a top club that has won a lot of trophies in the past - and will do so in the future we hope as well.
"I've got the trust of the manager and the club, which for me is very nice and I'm proud to be here."
Rodgers added that the 21-year-old, who will wear the number 29 shirt, is a player who is destined for big things in the future.
"He's technically strong," he's a good finisher with both feet, he can play central or on the sides in a 4-3-3 up front, he's quick, and he's tactically very good.
"He is arguably the best physical player I've worked with in terms of his pace, power and fitness.
"Mentally, he's very strong. We've got a player who is on the up and someone who I think Liverpool will benefit from."