Abraham smashed home a powerful header in the 11th minute at a bouncing Stadio Olimpico in Rome to put Jose Mourinho's team into their first European final in more than three decades.
They await Feyenoord in the Albanian capital Tirana on May 25 after the Dutch side prevailed 3-2 in the other semifinal, and they will be backed by an army of supporters who got behind their team in stirring fashion.
After the match, Roma announced tickets for the final would be distributed for free to season ticket holders, a reward for the phenomenal support shown to a team which currently sits fifth in Serie A.
"Before the game started, the fans won us this game," Abraham told BT Sport after netting his 25th goal of the season.
"We weren't even on the pitch and we heard them. The win was for them and my team. Hopefully, we can go all the way."
Roma fans have not seen their team win a trophy since 2008 and the roar let out at the final whistle was one of massive relief after spending much of the match on the back foot against the Foxes, themselves gunning for a first major continental final.
Brendan Rodgers' team ran into the ground against a typically well-organised Mourinho side who were content to let Leicester do the running and try to pick the away side off on the break once their aggregate advantage was established early on.
"They were better than us in the first half even if they didn't create much," Rodgers told reporters after the match.
"We had a lot of the play in the second half but we couldn’t make the breakthrough. Our Achilles heel all season has been set-pieces and it was again tonight."