The kidnappers' announcement was contained in a videotape aired on Thursday on Arab television.
The kidnappers, who identified themselves as Jundu Allah (Soldiers of God), claimed the killing was in response to Iran's "violations against Sunni figures in a number of Iranian towns," the video said.
A group of four armed men appeared standing behind a kneeling hostage as one of them read a statement threatening to kill the other captives.
The Iranian authorities confirmed that a group calling itself Jund Allah had admitted kidnapping nine Iranian soldiers near the Pakistani border.
But an Iranian intelligence ministry statement described the kidnappers as a local bandit group and said the hostages had been taken across the border into Pakistan's unruly Baluchistan province.