They become the first Ukrainian club to reach the last-eight of the competition in 12 years after beating the Italians 6-2 on aggregate.
Shakhtar Donetsk humble Roma to reach Champions League quarter-finals
Shakhtar Donetsk 3 // AS Roma 0 (Shakhtar win 6-2 on aggregate)
DONETSK, UKRAINE // Shakhtar Donetsk reached the Champions League quarter-finals for the first time when they humbled AS Roma 3-0 to complete a crushing 6-2 aggregate win.
Roma had gone into the match in buoyant mood under Vincenzo Montella, their new manager, 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.
The Italians were already ruing a missed chance after striker Marco Borriello fluffed a 28th-minute penalty.
The win makes Shakhtar the first Ukrainian side to reach the Champions League last-eight since Dynamo Kiev in 1999.
Tomas Hubschman set Shakhtar, the five-time Ukrainian champions, on the way to victory with a neat back flick from a cross by Willian in the 18th minute.
The 22-year-old Brazilian scored himself after the break, sidestepping David Pizarro and curling a shot in off the post.
Striker Eduardo scored the third in the 87th minute to complete the drubbing.
Borriello, the most dangerous man on the pitch for much of the game, had looked full of confidence as he ran up to take the first-half spot kick but he struck the ball weakly and Shakhtar goalkeeper Andriy Pyatov bundled it round the post.
Howard Webb, the English referee, 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.