The playmaker complained of pain following the 1-0 weekend win over Lecce and is set to spend three weeks on the sidelines.
Mkhitaryan, who joined on loan from Arsenal last month, looks likely to miss at least three club matches, including the Europa League trip to Austrian side Wolfsberger on Thursday.
The injury also robs Armenia of its captain for the upcoming Euro 2020 qualifiers against Liechtenstein and Finland.
Mkhitaryan made his fourth Serie A appearance and set up Edin Dzeko's winner as Roma earned three points at Lecce.
The victory has proved a costly one, with midfielder Lorenzo Pellegrini fracturing a metatarsal in his right foot during the same game.