The 20 year-old, who had been attracting interest from Manchester City, has penned a five-year deal and could cost the capital club a further $A2.2m in bonus payments.
Under, who has four Turkey caps to his name, now wants to establish himself in Eusebio Di Francesco's first-team plans.
"I'm delighted to be here and very excited about meeting the fans," he told the club's official website.
"My aim is to earn myself a regular starting berth as quickly as possible."
He is the fifth pre-season signing for Roma, who finished second last term and are strengthening under sporting director Monchi's guidance.
"We are extremely happy with the arrival of Cengiz, a player with great technical ability," said Monchi. "We are confident he will quickly adjust to his new surroundings and his quality and talent will be a valuable addition to our squad."
Roma have also brought in the likes of Hector Moreno, Maxime Gonalons and Juan Jesus.