The Ukrainian side pushed to score a vital equalizer, but their efforts appeared to be in vain.
That sending-off led to tensions spilling over, with Shakhtar's Facundo Ferreyra picking up a booking for shoving a ballboy over an advertising board. The incident led to a scuffle on the pitch, but Ferreyra was not penalized for his actions.
Cengiz Under's goal in the first leg last month ultimately proved crucial for Roma, but Dzeko was the hero on the night and also proved instrumental in Shakhtar defender Ordets being shown a straight red card with 12 minutes left.
Roma hadn't conceded in its three Champions League matches at the Stadio Olimpico this season but almost netted an own goal when Alessandro Florenzi tried to head clear a free kick but sent the ball flashing across the face of goal.
"We managed to do something that had been missing for 10 years", said Dzeko.
"It is not for me, but for everyone, we are deservedly in the quarter-finals", said the Bosnian after his fourth Champions League goal, and 18th in total for the club this season.
The Champions League round of 16 completes on Wednesday, and the draw for the quarterfinal round will take place on Friday, with no restrictions placed on which teams can be paired together.
Roma advance into the last eight of the competition, but Shakhtar's journey is over having been dumped out on away goals in Italy.
The Daily Mail recently reported that Fonseca is Farhad Moshiri's top target to replace Sam Allardyce as Everton manager this summer, and Toffees fans unsurprisingly had comments to make after Shakhtar's exit.
Spring Ahead Weekend is HERE!
Previously, Canada had observed Daylight Saving Time from the first Sunday in April until the last Sunday in October. People do not like dark mornings and thats the main reason daylight saving time has not been adopted year-round".
Amid lull, Red Cross delivers aid to Syria's Ghouta
More than 340,000 people have been killed since Syria's war started in 2011 with a brutal crackdown on anti-government protests. The U.N. said airstrikes and shelling in eastern Ghouta continued for hours while the convoy was unloading supplies.
New York Giants: Nate Solder free agent signing grade
Last night a small piece of good news was delivered: The Patriots will re-sign core special teamer Nate Ebner to a two-year deal. As reported by NFL Network's Ian Rapoport , Solder's deal with NY will make him the highest paid offensive lineman in the NFL.
Joy Ride with Mario on the streets of Google Maps
Nintendo and Google partnered to add Mario Kart to the search engine's map and directions app on Android and iPhone. Then simply spend the rest of your day wandering aimlessly siply so you can see Mario drive around your local area.