According to inscriptions from 1315 and 1341 the church is called St. Astvatsatsin, but is traditionally called "Mashtots Hayrapet" after the Armenian Catholicos Mashtots Yeghivardetsi, who died in Garni in 897. His gravestone is inside the "Mashtots Hayrapet" chapel (yes same name) near the site of the temple. In the 14th century it was called Sumpents monastery.
The church was built in Garni in the 12th century on what was the site of a pagan shrine.