The Ukraine outfit had a slight advantage heading into the clash after defeating the Serie A out 2-1 in the first leg.
Roma defender Alessandro Florenzi nearly netted an own goal in the 11th minute, but he was lucky to witness his miscued clearance sail inches past the upright.
In the 29th minute, Taison almost gave the Ukrainian side the lead with a well struck shot from 25-yards but it could not dip in time to beat Roma goalkeeper Alisson.
Clear cut chances were hard to come by for both sides who both resorted to long range efforts, none of which were threatening.
Seven minutes into the second half, Edin Dzeko broke the deadlock for Roma with a low strike inside the box following a through ball from Kevin Strootman.
The Italian side were in command of the match after the hour and continued to push for a second goal.
However, with 20 minutes remaining they started to retreat back and play on the counter-attack as the Ukrainian side upped their tempo looking for the equaliser.
The visitors suffered a big blow ten minutes before time as Ivan Ordets was sent off after pulling down Dzeko who was clear on goal.
Nevertheless, Roma remained resolute in the closing to see out the win and book their spot in the next round.
The Italian side booked their spot in the last eight on the away goals rule, with the tie ending 2-2 on aggregate.