The Italy youngsters both scored inside the opening 13 minutes as Rudi Garcia's men made it three wins from three this season to keep pace with Juventus
Destro opened the scoring after just 10 minutes of play and Florenzi added a second moments later, effectively ending the game as a meaningful contest.
Zeman had not faced his former club since being sacked by the Giallorossi two years ago and had arrived in Rome insisting that he was not looking for revenge, which was just as well, really.
Cagliari were unbeaten in their previous four visits to the Olimpico but they folded inside 15 minutes this time around.
Destro had already tested Alessio Cragno with a low shot when he broke the deadlock he tapped home a Florenzi cross at the far post.
The latter then turned goalscorer. Gervinho controlled a long punt forward into the area and although the Ivorian's shot was blocked, the ball broke for Florenzi.
The Italy international finished emphatically first time with his right foot and then ran into the stands to hug his grandmother, who had come to see him play at the Stadio Olimpico for the first time.
Florenzi's 82-year-old nan was visibly moved by the gesture but the referee Sebastiano Peruzzo didn't see the funny side and promptly booked the midfielder.
Roma went close to extending their lead, with Maicon seeing a piledriver fly just wide and Gervinho failing to beat Cragno, who was making his Serie A debut, with a close-range strike.
In the end, Rudi Garcia's men did not manage to add to their tally but their early two-goal salvo proved sufficient to earn the Giallorossi a third successive Serie A win of the season.
The one negative on an otherwise positive afternoon, though, was the injury that forced Daniele De Rossi off with just eight minutes to go.