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Goals from Pedro, Edin Dzeko (2), Jordan Veretout and substitute Carles Perez proved enough for the Giallorossi, who went behind early on and were pegged back at 2-2 just minutes into the second half.
But those two setbacks - both of them unfortunate in their own way - would not derail Paulo Fonseca's side, who were eventually rewarded for all the chances they created.
It was Pedro - with his first goal at the Olimpico - who first erased Roma's deficit, as he stroked home Lorenzo Pellegrini's pass inside the box.
Then it was Dzeko - moving fourth alone on the club's scoring charts - who finished off a sweeping counter-attacking move to give the team the lead at the break, even if it was cancelled out soon after the interval as Gianluca Lapadula followed in his saved penalty to equalise.
Pedro responded by winning a penalty of his own - converted by Jordan Veretout - before Dzeko tapped in again to effectively wrap up the three points.
In the dying moments, however, Carles Perez made a slaloming run before producing an arrowed finish to add an extra gloss to the scoreline.
Fonseca's side are back in action on Thursday - they face Young Boys in Switzerland in the first game of the Europa League group stage.
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