Abraham finished the 2020-21 season as Chelsea's joint top scorer with 12 goals but saw relatively limited playing time after Thomas Tuchel replaced Frank Lampard.
"You can sense when a club really wants you -- and Roma made their interest clear immediately," Abraham said.
"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 No. 9 at this club and I just can't wait to get started and to help the team."
Chelsea manager Thomas Tuchel had said at a news conference last Wednesday that Abraham was not guaranteed regular starting time at Stamford Bridge this season.
"I can absolutely understand that he wants more minutes," Tuchel said.
"So the decision will be 'how do we plan, what are Tammy's plans, does he fight his way back into the team from the position where he ended last season or does he want to change club to have the chance of being a regular starter?'"
Chelsea signed Romelu Lukaku last Thursday which added to speculation that Abraham would move away from the club, and he was greeted by several Roma fans when he landed at the city's Ciampino airport on Sunday.
"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," Tiago Pinto, Roma's General Manager said.
"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."