The first chance of the game fell to the visitors, who saw Carrasco rifle an effort into the hands of Soslan Dzhanaev from a free kick two minutes in.
Moments later, Angel Correa pounced on a rebounded ball but blasted his chance several inches wide of the right-hand post.
The hosts struggled to get into the game but held firm until just after the hour, when Carrasco pounced onto a rebound from Juanfran's effort and stabbed home.
Antoine Griezmann then nearly doubled Atletico's lead seven minutes from time, but Dzhanaev leapt acrobatically to keep out the Frenchman's header that was seemingly destined for the top-left corner.
The result means that last season's Champions League finalists maintain their perfect start at the top of Group D, while Rostov, who lose their 26-game unbeaten run at home, remain third.