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."
Maxi Rodriguez has left Liverpool to rejoin Newell’s Old Boys in his native Argentina after two-and-a-half years at Anfield.
Winger-forward Maxi, 31, arrived at Liverpool from Atletico Madrid in January 2012, making 73 appearances and scoring 17 goals.
He fell out of favour under former boss Kenny Dalglish, with new manager Brendan Rodgers hinting that the Argentine player and his family were struggling with life in England.
"Liverpool Football Club can today confirm that Maxi Rodriguez has joined Newell's Old Boys in Argentina," Liverpool said on their website.
"Everyone at Liverpool FC would like to thank Maxi for his contribution to the club and wish him all the best for the future."
Maxi left Newell’s in 2002 to join Spanish club Espanyol, from where he moved to Atletico in 2005.
He became skipper at the Vicente Calderon but was allowed to leave before the 2010 World Cup, then becoming a fan favourite at Anfield.
Rodgers had said earlier on Friday that conversations with Maxi led him to believe he and his family were keen on returning to Argentina, but this move comes quicker than anyone could have anticipated.