Abraham has signed a five-year contract with the Serie A club to fill the void left by Edin Dzeko's departure to Inter.
Roma paid €40million (£34.1m) to secure the services of a player who made his Chelsea debut in 2016 and has played six times for his country.
Mourinho believes leaving the Premier League behind for a new challenge shows Abraham has lofty aspirations.
"With Tammy, I prefer to say, 'wait and see'," Mourinho said in a video posted on Roma's Twitter account.
"I say that with total confidence. I've known him since he was a boy. He's never played for me because he was a 14, 15, 16-year-old boy when I was at Chelsea, but I know him very well.
"I know him as a player, a person and in terms of his mentality. I know how he made the decision to leave the Premier League, which is always tough for an English player.
"That tells me so much about him, because when you leave the Premier League, you do so because you're ambitious.
"You leave because you want to get back in your national team, because you want to play at the World Cup, because you want success outside of England, where not many English players have had brilliant careers.
"He comes here with that ambition and we hope to see his best qualities as a player."
Chelsea will reportedly receive a further €5m in add-ons and have an option to re-sign the 23-year-old, who fell further down the pecking order when Romelu Lukaku returned to Stamford Bridge.
After scoring 15 Premier League goals for the Blues in the 2020-21 season when Frank Lampard was in charge, Abraham saw his opportunities diminish under Thomas Tuchel last term.
The 23-year-old started only three league fixtures following the appointment of Tuchel in late January, while he did not feature at all in the knockout stages of the Champions League as Chelsea went on to lift the trophy in Porto.
He also missed out on England's squad for Euro 2020, with Dominic Calvert-Lewin preferred by Gareth Southgate as a direct back-up to Harry Kane.
Abraham will be hoping to have similar success in Serie A as another England international, Chris Smalling, who rejuvenated his career with a loan switch to Roma in 2019-20, which was made permanent last year.