Paulo Fonseca's men came to Stadio Olimpico in Rome, hoping to keep their goals safe from Roma's attacks, which would let them qualify into the quarterfinals.
Edin Dzeko credited his Roma team-mates for a "patient" performance as they dumped Shakhtar Donetsk out of the Champions League.
Ukrainians Shakhtar had Ivan Ordets sent off after he pulled Dzeko down by the shirt.
Having trailed 2-1 from the first leg, Roma looked bereft of ideas for much of the contest before Dzeko broke the deadlock in the 52nd minute following a tepid first half devoid of goalmouth action. The Bosnian took a defence-splitting pass from Kevin Strootman into his stride and beat the offside trap before he steered the ball into the far corner with the outside of his foot, past the advancing Pyatov.
"We were patient. We knew that one goal could see us reach the quarter-finals".
"I'm proud of this team because I think we knocked out a very strong team".
Shakhtar were unlucky not to get an away goal of their own early on when Roma's Alessandro Florenzi nearly headed an in-swinging free-kick into his own net with a glancing clearance.
Roma's solitary glimpse of goal in the opening half came in the 28th minute when the visiting goalkeeper, Andriy Pyatov, palmed away a fizzing low cross by Aleksandar Kolarov to deny Dzeko a simple tap-in from close range.
The home side came out with renewed objective in the second half and Dzeko delighted their vocal fans when he broke the deadlock with a trademark clinical finish. Shakhtar goes home, though.
Roma's former captain Francesco Totti, who watched anxiously from the terraces, was overjoyed at fulltime. "Among the eight best sides in Europe and we push onto the next round".