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The English’s striker’s 27th goal of the season proved to be the only goal on the night, and that was enough for Jose Mourinho’s side to go through after a 1-1 draw a week ago at the King Power Stadium.
The Foxes fell short in their first European semi-final and there will be a lingering sense of regret they failed to seriously trouble Roma and never did themselves justice.
The club have torn up the rule book after winning the Premier League and FA Cup in the last six years but, this season, European success was just out of reach.
A pre-match banner declared ‘Everyone in Britain dreads the name of the Romans’ and Roma’s determination was obvious.
Leicester did have early protests for a penalty ignored when Chris Smalling hauled down Wesley Fofana but the hosts quickly took charge to grab an 11th-minute lead.
Kasper Schmeichel had already saved Lorenzo Pellegrini’s cute free-kick as Roma forced a number of corners and it was, again, from a set-piece where Leicester conceded.
Pellegrini’s dangerous delivery was met by Abraham and he outjumped Ricardo Pereira to power a towering header past Schmeichel from seven yards.
It was the striker’s ninth goal in the competition this season - with Alan Shearer and Stan Bowles the only Englishmen to score more in a single European campaign.
Shell-shocked, Leicester retreated and Pellegrini, scorer of Roma’s first-leg goal, was a constant menace with Jonny Evans mopping up after Schmeichel saved another effort.
Boss Brendan Rodgers had spoken about using the 72,000 sell-out crowd to Leicester’s advantage, hoping to play on any nerves and anxiety with Roma so close to ending a 14-year trophy drought.
But his side were never able to find sustained momentum to even get under the hosts’ skin on the pitch.
Rodgers rolled the dice at the break with the ineffective Ademola Lookman and Harvey Barnes replaced by Kelechi Iheanacho and Daniel Amartey.
A double change at the break helped the Foxes overcome a second-leg deficit to beat PSV in last month’s quarter-final and immediately they were more robust but still lacked the imagination to level the tie.
Jamie Vardy did have a shot blocked by Abraham and the Foxes saw more of the ball, with James Maddison curling at Rui Patricio with 13 minutes left.
But a low-key second half suited Roma and, even though they created little themselves, they were always comfortable.
Iheanacho shot at Patricio from distance as time slipped away and, this time, there was no fairy-tale ending for the Foxes.
Additional reporting by PA Sport.