Alkhanov said on Chechen State Television on Monday that the gunfire apparently came from an armoured personnel carrier assigned to a checkpoint between the village of Kurchaloi and Geldagan.
The attack happened near the checkpoint on Friday, he said.
"It does not matter to the population of the republic of Chechnya who has perpetrated this crime - guerrillas or people wearing shoulder-straps," the Kremlin-backed Alkhanov said in televised remarks.
"Crimes committed by the latter deliver a heavier blow to the Chechen people because they hope for their protection."
Alkhanov said an investigation is under way. Major-General Ilya Shabalkin, spokesman for the federal forces in Chechnya, refused to comment.
The Russian military's heavy-handed approach to Chechnya has alienated much of the local population and stoked tension between the military and Chechen officials.
Rights groups say killings of Chechen civilians by Russian soldiers are common, and they have called for more prosecutions. The Russian military insists that it is prosecuting abuses.
A armoured personnel carrier
allegedly opened fire on the car
Alkhanov told Chechen State Television that the armed men did not ask the passengers to produce any identification or make any other demands, although it was not immediately clear how he would know that.
Russia's NTV television broadcast footage of bloodied bodies lying in brush near the roadside.
Meanwhile, Shabalkin said earlier Monday that Russian forces killed two rebels and prevented several rebel bands from leaving Chechnya for neighbouring Dagestan on a mission to carry out "terrorist acts".
He said the rebels had been cut off in mountainous districts of Chechnya after federal forces received information that rebels close to rebel leaders Shamil Basayev, Aslan Maskhadov and Doku Umarov were ordered to secretly cross the Russian border and carry out attacks outside Chechnya, probably in neighbouring countries.
One rebel was killed in the Vedeno region, and another in the Itum-Kale region, both of which border Dagestan, Shabalkin said.
"Crimes committed by [those wearing shoulder straps] deliver a heavier blow to the Chechen people because they hope for their protection"
Chechen President Alu Alkhanov
Earlier this month, Chechen rebel leader Aslan Maskhadov announced a ceasefire that was to last until Wednesday, the anniversary of the brutal, Stalin-era deportation of Chechens and Ingush en masse to the then-Soviet republic of Kazakhstan.
Russian military officials insist, however, that rebel attacks have continued in spite of the truce, which they have dismissed as a publicity stunt, and they contend that most of the rebels are beyond Maskhadov's control.
Russian troops withdrew from Chechnya in 1996 after a disastrous 20-month war that left the region de facto independent.
Troops rolled back in 1999 after rebels raided the neighbouring southern Russian region of Dagestan and after a series of apartment house explosions in three Russian cities that authorities blamed on the rebels.