Needing a win to confirm their spot in the last-16, Diego Perotti's header moments half-time proved to be enough for the Giallorossi - who ended up finishing top of their group on head-to-head record, after Chelsea drew 1-1 at home to Atletico Madrid.
That capped off what had already become a momentous day for the club, with news filtering through barely an hour before kick off that the club had been granted final approval for their proposed new stadium, the Stadio della Roma.
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Eusebio Di Francesco's side will now be seeded for Monday's draw, where they could face teams including Real Madrid, Bayern Munich and FC Basel.
In truth it was a somewhat nervy 90 minutes out on the pitch, with Qarabag - who drew both games with Atletio Madrid - happy to defend deep.
Roma pushed and probed but ultimately it came down to Perotti to grab another vital goal - as he headed home from Edin Dzeko's saved shot to change the mood inside the Olimpico.
That put Roma on 11 points, a mark Chelsea would join them on after they equalised at Stamford Bridge. But the Giallorossi's 3-0 win over the Blues at the Olimpico (and 3-3 draw in London) was enough for them to edge the head-to-head tiebreaker.
Di Francesco restored both Radja Nainggolan and Diego Perotti to the starting line-up after they missed Monday's game through minor injuries - while Daniele De Rossi resumed the captaincy.
In defence, Federico Fazio partnered Kostas Manolas.
14' - Chance for the hosts. Dzeko wins a corner, which is taken short. El Shaarawy eventually fizzes in a low ball that Manolas flicks towards goal - but Sehic makes a comfortable save.
17' - NO! Another corner sees the team bring out a training ground routine: De Rossi dummies Kolarov's low delivery but Dzeko cannot put his free shot on target!
25' - El Shaarawy beats the onrushing goalkeeper to ball out wide on the right, but his ball into the box isn't on target and no-one is there to turn it home!
31' - Some slick interplay from Roma before Strootman chips the ball in for El Shaarawy - but he volleys over!
32' - Richard Almeida shoots wide from 20 yards as Qarabag launch another quick counter. Alisson did not look worried.
40' - CLOSE! Perotti's dangerous inswinging free-kick is glanced on by Manolas, but the touch isn't enough and the ball flies just beyond the far post!
45' - CLOSE AGAIN! Radja's thumping low drive is kept out by Sehic and the rebound is eventually gathered!
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48' - Radja hits a low drive that deflects off the defender on the way through to the goalkeeper.
53' - GOAL! The deadlock is broken! Perotti dribbles into the box and feeds Dzeko for the chance. Sehic saves his shot... but Perotti follows in the rebound and heads home from a tight angle! DAJE!! 1-0!
60' - Plenty of pressure from the Giallorossi now. Defenders put their bodies on the line to deny Dzeko and Nainggolan in quick succession.
62' - Dzeko and Yunuszada are booked for tussling in the box at the corner. When the ball eventually comes in, Radja collects a half-clearance and forces a good stop from Sehic!
79' - A Kolarov free-kick goes narrowly wide.
83' - Big, big chance for Qarabag. Ndlovu wriggles free on the left and puts in a great cross that Michel heads straight at Alisson.
86' - Kolarov beats two defenders at the byline and then hits a powerful low strike that Sehic has to tip wide!
90' - Roma counter quickly through Gerson, who passes to Dzeko... but he drags his shot wide!
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“It's a fantastic moment for us. When the draw was made no-one really believed in us, but we got our heads down and worked really hard. We deserved to finish top.”
- Diego Perotti
“It was a great win and a difficult one too – it's never easy to win in the Champions League. I'm absolutely delighted to finish top of the group. It was tough. No one had us down as favourites at the start; in fact, we were pretty much written off. We just got on with the job and believed in ourselves."The lads played with real conviction right from our opening game at home against Atletico. We didn't play particularly well that day but we showed we could dig in. That was the start of a great group stage for us.”
- Eusebio Di Francesco
The draw for the last-16 takes place on Monday 11 December.
The Giallorossi are back in Serie A action on Sunday, as they travel to face Chievo Verona in the day's early kick-off.
Goalscorers: Perotti (53')
Roma: Alisson; Florenzi, Manolas, Fazio, Kolarov; De Rossi, Nainggolan, Strootman; Perotti (Pellegrini), Dzeko, El Shaarawy (Gerson).
Qarabag: Sehic, Yunuszade, Rzezniczak, Medvedev, Guerrier, Garayev, Richard Almeida, Michel, Izmayilov, Madatov, Ndlovu.
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