Giallorossi legend Totti left this summer after becoming fed-up with his front office role, while De Rossi is now a Boca Juniors player after his contract was not renewed.
"Something's been lost. They showed personality, on and off the pitch," explained the 33-year-old.
"Now it's up to me, [Federico] Fazio, [Alexsandar] Kolarov, [Alessandro] Florenzi and [Antonio] Mirante to guide the younger players."
Dzeko has been captain several times this season and he added: "It's a armband that weighs on you. Players like Totti and De Rossi wore it in the past. But I'm ready."