The first explosion hit a convoy of cars carrying pro-Moscow police at 7.30am (0430 GMT).
The second hit the main city hospital 45 minutes later as the wounded were brought in, injuring civilians including a doctor. Both explosions took place on Sunday.
The powerful second bomb smashed windows at the hospital, destroying the car that carried it and damaging others.
"Some people are severely injured, but there is no threat to their lives. All 13 are in the ninth city hospital, where the explosion took place," said Ilya Shabalkin, spokesman for the Russian military command in Chechnya, where Russia has been fighting separatists for a decade.
Rebels frequently lay mines on the roads of the mountainous region, targeting military and police convoys. Russian soldiers are killed almost daily, and automatic gunfire echoes around Grozny and other towns at night.
A member of the Kadyrovtsy - a pro-Moscow paramilitary force made up mainly of former rebels and led by the son of assassinated local leader Akhmad Kadyrov - said the Sunday bombings were part of a feud with rebels.
"We are battling them, and this feud does not stop here," said the heavily armed man, who identified himself only by his nickname, Bely, as he stood outside the hospital.