The aggregate score ended level at 2-2, with Jimmy Floyd Hasselbaink's strike in the Stadio Olimpico proving crucial in a game that saw seven yellow cards dished out, as well as a red to home defender Philippe Mexes.
The home side dominated the early proceedings as they pushed to claw back Boro's one-goal advantage, with Cesare Bovo at the heart of most of their good work.
Bovo came within inches of netting after 24 minutes as his 25-yard effort cannoned back off the crossbar with Mark Schwarzer well beaten.
Just past the half-hour mark Mancini escaped with a caution for a crude elbow on Gaizka Mendieta, but Boro exacted revenge immediately as Stewart Downing's cross from the left was headed home by Hasselbaink at the near post.
However, the lead lasted just 11 minutes before a sweeping Roma move ended with Rodrigo Taddei pulling the ball back for Mancini to convert beyond a stranded Schwarzer.
Straight after the restart Schwarzer denied Mancini from close range and then produced a fantastic stop to deny Daniele De Rossi a seemingly certain goal.
However, Roma did take the lead on the night with 24 minutes left to play after Ray Parlour brought down Stefano Chuka Okaka and Mancini converted from the penalty spot.
The home side then poured forward in search of a third goal that would have sent them through, but Boro held firm to seal a place in the last eight of the competition.
Roma's misery was completed late on as France international centre-half Mexes was dismissed after two bookings.