The England international headed home after just 11 minutes to break the deadlock at the Stadio Olimpico, a goal that would ultimately prove decisive both on the night and in the tie.
Roma will now face Dutch side Feyenoord in the final in Tirana, which will take place on May 25.
With the tie poised at 1-1 after last week's first leg in England, it was the Giallorossi who started the brighter - with 64,000 fans inside the Olimpico roaring them on.
Crucially, like in similarly important home games this season against Lazio and Bodo/Glimt, they took advantage of that pressure and scored an early goal - and, just as in those games as well, it was Abraham who got it.
This time the No. 9 rose highest from the corner to head in Lorenzo Pellegrini's delivery, sending the crowd into rapture with 11 minutes on the clock.
That goal put the onus on Leicester to find an equaliser, but Brendan Rodgers side struggled throughout to make any clear-cut openings - as Rui Patricio dealt comfortably with the limited chances that came his way.
At the other end the Giallorosse were only fleetingly threatening, with Nicola Zalewski almost finding an all-important second goal.
In the end, however, it wasn't necessary; as Mourinho's men saw out five minutes of added time with relative ease to book their place in the final in Tirana.
ROMA 1-0 LEICESTER CITY
(Roma win 2-1 on aggregate)
GOALSCORER: Abraham (11')
ROMA (3-5-2): Rui Patricio; Mancini, Smalling, Ibanez; Karsdorp, Sergio Oliveira, Cristante, Pellegrini Zalewski (Vina); Zaniolo (Veretout), Abraham (Shomurodov)
LEICESTER (4-3-3): Schmeichel; Pereira (Castagne), Fofana, Evans, Justin; Maddison, Tielemans, Dewsbury-Hall (Perez); Lookman (Amartey), Vardy, Barnes (Iheanacho)
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