Roma had gone into the match in buoyant mood under new manager Vincenzo Montella but the game was virtually over when, trailing 1-0, they were reduced to 10 men just before half-time when Philippe Mexes was sent off for a second yellow card.
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"We played much better than they did. Of course, 11 against 10 is easier. They were nervous and made a lot of mistakes and we knew how to control the situation," said Shakhtar coach Mircea Lucescu.
The 22-year-old Brazilian scored himself after the break. Dario Srna's corner was only half cleared and Willian curled the ball into the far corner from the edge of the penalty area.
Striker Eduardo scored the third in the 87th minute to complete the rout by the side owned by Ukraine's richest man, magnate Rinat Akhmetov.
British referee Howard Webb had no hesitation giving Mexes, Roma's central defender, a second yellow card for a clumsy foul on Luiz Adriano, reducing Roma to 10 men for the second half.
Montella said Roma would learn from the defeat. "We have to develop the ideas and the pace that I saw in the first 30 minutes of the game.
"I tried to keep up the competitive spirit of the team despite the nervous atmosphere after Mexes was sent off. We will analyse our mistakes and move forward."
The coach said he would keep faith with Borriello. "Though he missed a penalty Borriello will stay at Roma, I count on him."
