The 20-year-old, who had also been linked with Manchester City, has arrived at the Giallorossi for an initial €13.4m fee, which could eventually rise by an additional €1.5m in add-ons.
Cengiz becomes Roma's fifth summer signing under transfer guru Monchi, following Hector Moreno, Rick Karsdorp, Lorenzo Pellegrini and Maxime Gonalons in joining the club.
And speaking to the Serie A side's website, Cengiz said: “I'm delighted to be here and I'm very excited about meeting the fans.
“My aim is to earn myself a regular starting berth as quickly as possible."
Sporting director Monchi added: "We are extremely happy with the arrival of Cengiz, a player with great technical ability.
"We are confident he will quickly adjust to his new surroundings and his quality and talent will be a valuable addition to our squad."
Cengiz scored seven goals and registered five assists in 32 Super Lig games last season as Basaksehir finished second in the Turkish top flight.
The forward made his debut for the Turkey senior team against Kosovo last November before scoring his first goal for his country against Moldova in March this year.