Trailing 2-0 after the first leg in the semi-final clash, the Giallorossi were always facing an uphill battle - one they made even harder for themselves when Sergej Milinkovic-Savic broke the deadlock just before half-time.
That left Luciano Spalletti's side needing to score four times to progress, something that looked unlikely even when Stephan El Shaarawy equalised almost immediately.
In the second half Roma's attempted charge was halted by a counter-attacking goal from Ciro Immobile. Mohamed Salah then scored twice in the final 20 minutes to put Roma ahead on the night, but their aggregate exit had already been effectively confirmed.
In the end the one positive from the night was the return of the Curva Sud, who roared on their team for 90 minutes despite the odds being so stacked against them.
3': Almost a perfect start. Emerson finds Dzeko in the box but his first-time effort is just wide. Dangerous!
20' - Close! El Shaarawy forces a save from Strakosha, who has to get up quickly to gather Dzeko's subsequent header!
37' - Goal. Lazio score on the counter. Immobile forces a good save from Alisson but Milinkovic pokes in the rebound. 0-1.
43' - GOAL! That's one back! Rudiger gets to the byline, the defender misses his clearance at the near post, and El Shaarawy is there to sweep home! 1-1!
See more images in our dedicated match gallery.
56' - Goal. Lazio back ahead. Immobile breaks the offside trap and converts beyond Alisson. 1-2.
66' - GOAL! Roma equalise again - Salah rifles home the rebound after El Shaarawy turns inside and hits the post. 2-2.
73' - Strootman hits a pure strike from 20 yards but Strakosha makes a good save.
82' - CLOSE! Salah gets a good look at goal inside the box but Strakosha blocks away his first-time strike.
85' - How did that not go in? Strakosha flaps at a punch but Strootman's follow-up strike is somehow deflected over...
90' - GOAL. Another for Salah - he gives Roma the lead on the night. Four minutes of added time. 3-2.
See match statistics in our dedicated match centre.
Roma's next game is in Serie A, as they travel to face Bologna on Sunday afternoon.
The following Saturday they host Atalanta at the Stadio Olimpico.
(Lazio win 4-3 on aggregate)
Goalscorers: El Shaarawy (43'), Salah (71', 90'); Milinkovic (37'), Immobile (56')
Roma: Alisson; Emerson, Jesus (Peres), Manolas, Rudiger; Paredes (Totti), Nainggolan, Strootman; El Shaarawy (Perotti), Dzeko, Salah.
Lazio: Strakosha; Bastos, de Vrij (Hoedt), Wallace; Basta, Milinkovic, Biglia (Murgia), Lulic, Lukaku; Felipe Anderson (Keita); Immobile
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