With the struggling hosts playing without pressure, the Giallorossi were forced to toil away for much of the contest at the Stadio Ezio Scida - but in the end a couple of clinical moments were enough to deliver the expected victory.
It was Nainggolan - wearing the armband for the day - who broke the deadlock, adding to his fine goalscoring run in recent weeks. He latched onto Mohamed Salah's short pass before turning a lashing a low shot beyond Crotone goalkeeper Alex Cordaz, finally breaking the deadlock just before the break.
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That strike will have been of particular relief to Dzeko, who had earlier missed the opportunity to open the scoring from the penalty spot - after Salah had been brought down.
The striker picked himself up and eventually decided the result with 10 minutes remaining, tapping home from six yards after Leandro Paredes had found Salah for a straightforward assist.
Crotone had their moments in attack - Wojciech Szczesny made one or two saves of note, while Diego Falcinelli had a goal ruled out for offside - but Roma were never unduly worried on their way to the final whistle.
League victory secured, attention now turns to the return of Europa League action - and this week's visit to Spanish side Villarreal.
Coach Luciano Spalletti opted to put Mohamed Salah straight into the starting line-up for his first slice of Giallorossi action since return from the Africa Cup of Nations.
He replaced Stephan El Shaarawy in the attacking lineup, while the other notable change saw Leandro Paredes come into the midfield in the place of Daniele De Rossi.
With De Rossi on the bench and Francesco Totti absent through illness, Radja Nainggolan assumed the captaincy.
17' - PENALTY ROMA! SALAH IS FOULED!
18' - Disappointment for the Giallorossi ... Dzeko drags his shot just wide of the left-hand post...
20' - Roma have responded quickly to that frustration, Emerson fires in a high cross that Salah nods narrowly over.
40' - GOAL! ROMA LEAD! Brilliant stuff from Nainggolan, who collects Salah's pass before turning and lashing a low shot beyond Cordaz! 0-1!
See more images in our match gallery.
51' - Dzeko takes a touch and hits a half-volley on the turn ... but he can't quite direct it on target.
54' - Roma hit the woodwork twice! First Dzeko's deflected shot clips the bar, then Fazio's header hits the post!
58' - Goal disallowed for Crotone. Falcinelli squeezes a shot in, but the flag was already up against him for offside.
60' - Szczesny saves well from Maxwell Acosty's speculative effort...
77' - GOAL! DZEKO DOUBLES THE LEAD! Paredes makes the opening with a sublime ball in behind the defence, and Salah squares to Dzeko for a straightforward finish! 0-2!
See more match statistics in our match centre.
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"I missed [the penalty] but it’s already out of my mind. I’ll let someone else take it next time. I’m pleased to have got a goal, but I always want to score more. The win is the only thing that matters."
- Edin Dzeko
“It was a difficult game, but we played the right way and kept the ball safe. But in the end we scored two goals – yes we could have scored more, with the penalty – but two goals, a clean sheet, very good.”
- Antonio Rudiger
The Giallorossi now enter a busy spell in their season, as they return to Europa League action with Thursday's last-32 first leg against Villarreal in Spain.
The weekend sees them host Torino in Serie A action at the Stadio Olimpico - which will then also be the venue for the following Thursday's return meeting with Villarreal.
Goalscorers: Nainggolan (40'), Dzeko (77')
Crotone: Cordaz; Djamel, G. Ferrari, Dussenne, Ceccherini, Rosi; Tonev, Crisetig, Capezzi, Boadu Maxwell; Falcinelli.
Roma: Szczesny; Manolas, Fazio, Rudiger; Peres (Rui), Strootman, Paredes, Emerson; Salah (De Rossi), Nainggolan (Perotti); Dzeko.
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