The England forward has signed a five-year deal with the Serie A outfit and departs Stamford Bridge having scored 30 goals in 82 appearances after coming through the Blues academy.
Abraham, who will wear the number nine shirt, said on his new club's website: "You can sense when a club really wants you - and Roma made their interest clear immediately.
"Roma is a club that deserves to be fighting for titles and trophies. I've had the experience of winning major trophies and I want to be in those competitions again - so I want to help this team to achieve that and get to the level where Roma should be.
"It's a massive honour to be the number nine at this club and I just can't wait to get started and to help the team."
Abraham moves on in search of regular first-team football and will work with ex-Chelsea manager Jose Mourinho at the Stadio Olimpico.
Former Blues boss Frank Lampard put his faith in the 23-year-old during the 2019-20 campaign, but the arrival of Thomas Tuchel in January saw the England international's chances limited.
Chelsea director Marina Granovskaia said on the club's website: "Tammy will always be welcomed back at Stamford Bridge as one of our own.
"We have all enjoyed watching his progress through our academy and into the first-team squad, and are grateful for his contribution to our successes at senior and youth level.
"His many goals in a Chelsea shirt and, of course, his part in our Champions League triumph last season will never be forgotten. I'm sure all Chelsea supporters will join me in wishing Tammy a long and successful career."
Roma are confident there is plenty more to come from Abraham, who had fallen further down the Chelsea pecking order following the £97.5m arrival of Romelu Lukaku from Inter Milan.
"Despite still being very young, with a huge amount of potential to keep improving, Tammy has already played more than 200 games in his career and scored over 100 goals - and won a number of major trophies too," Roma's general manager Tiago Pinto said on the club's website.
"Choosing to leave the Premier League, and the club he grew up at, demonstrates very clearly just how much he believes in this opportunity to develop his own game and show what he can do at Roma.
"Bringing in players that have such a hunger and desire to play for our club is hugely important in helping to build the identity and sense of belonging that is a fundamental part of our overall vision for this team."