On August 31, assailants broke into a house in Karabulak and shot dead a man and his two sons, aged 20 and 24 years, with an assault rifle. Their mother Vera Draganchuk, an ethnic Russian teacher, managed to escape.
The special operation took place on Sunday. Both militants are also suspected of carrying out attacks against civilians and army personnel, the press office said.
The murder of the three men is the latest in a series of similar crimes in the volatile republic, bordering on Chechnya, which saw two Moscow-led military campaigns against separatist regimes in the 1990s and early 2000s.
In the early hours of July 16, another ethnic Russian teacher, Lyudmila Terekhina, 55, and her son and daughter were shot in their house in the Ingushetian village of Ordzhonikidzevskaya. Policemen said possible motives were robbery and racial tension. At their funeral two days later, a tripwire bomb went off, leaving 11 people injured.