Kumbulla, 20, joins the club from Hellas Verona on an initial two-year loan deal that runs until 30 June 2022.
As part of the negotiations, two-year loan arrangements for players Mert Cetin, Matteo Cancellieri and Aboudramane Diaby to go in the opposite direction and join Verona have also been agreed.
When the costs of all loans are weighed against each other, the net cost to Roma is €2 million.
The overall transaction also includes various obligations, based on the meeting of certain targets, to make all four transfers permanent.
Assuming the four deals are all completed, the net cost to the Giallorossi will be €13.5m.
The deal for Kumbulla also includes a number of specific performance-related bonuses, based on the achievements of both the player and the club, that are worth up to a maximum of €3.5m.
Should the conditions be met to make the transfer permanent, Kumbulla has agreed a contract with the club that runs until 30 June 2025.
“Roma represents an opportunity, and a challenge, impossible to turn down,” said Kumbulla.
“I was convinced by the plan the club has in place and I had no hesitations about making this choice - I will do everything to ensure I don’t disappoint those who have shown their belief in me.”
A product of the Hellas Verona academy setup, Kumbulla made his debut for the club in August 2018.
Last season, as Verona returned to Serie A, he played 25 times in the league to help Ivan Juric’s side to an impressive top-half finish.
Born in Peschiera del Garda, near Verona, Kumbulla opted to represent the country of his parents at international level.
Having represented Albania at Under-17, Under-19 and Under-21 level, he made his senior debut for his country last year.
“Marash was one of our top targets and we are delighted to have signed him,” said club CEO Guido Fienga.
“With him, we are bringing to Roma a young talent who stood out last season – and a player who possesses all the attributes to form, alongside others, a key part of the group we are building for the future.”
Kumbulla may yet make his debut for the Giallorossi against his old club – Roma travel to face Hellas Verona in the first game of the 2020-21 Serie A season on Saturday.
He will wear the No. 24 shirt.
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