Claudio Ranieri’s side pulled two points clear of AC Milan, who face Lazio later on Saturday, in the final Champions League spot.
However, Milan and Atalanta, who host Empoli on Monday, will both leapfrog Roma in the standings if they win their matches over the next two days.
Dzeko struck midway through the second half to score his eighth league goal of the season, but his first in Rome since notching a brace against Chievo last April.
Roma are fourth on 54 points, with Milan and Atalanta on 52 while Lazio are on 49 points and have two games in hand over their city rivals.
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(Reporting by Alasdair Mackenzie, editing by Pritha Sarkar)
