The pair scored in the opening 15 minutes in the Azeri capital to give the Serie A side a two-goal lead but the visitors endured a nervy second half after Pedro Henrique pulled one back in the 28th minute.
The result left Roma top of the group on four points from two games, one ahead of Chelsea, who meet Atletico Madrid (one point) in Spain later on Wednesday.
Centre back Manolas fired Roma ahead with a diving close-range header in the seventh minute, scoring his first goal for the club in European competition after Lorenzo Pellegrini drove the ball into the six-yard box.
Dzeko made it 2-0 when he volleyed an angled shot off the underside of the bar past fellow Bosnian Ibrahim Sehic from eight metres, having chested down a looping pass by strike partner Stephan El Shaarawy.
Roma looked in complete control when Brazilian Henrique scored Qarabag's first goal in Europe's elite competition, beating goalkeeper Alisson Becker with the outside of his foot after a fine pass by Dino Ndlovu.
Sehic produced a string of superb saves as Roma missed several chances to put the game beyond Qarabag, who pressed forward with long balls in the closing stages but could not find a way through.
South African Ndlovu missed the home side's best chance of an equaliser in the 90th minute, when he headed a deep cross from the left inches wide of the near post.
(Writing by Zoran Milosavljevic; Editing by Toby Davis)
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