The bodies of 10 fighters were reportedly found after security forces pounded their positions using mortars and helicopter gunships. Four others were shot dead as they drove their car through Makhachkala late on Friday.
Vesti-24 showed four bodies lying among shattered glass near a battered, white Lada car.
Ramazanov said the men in the car had been involved in "explosions, assassination attempts and attacks on police officers".
Adilgirei Magomedtagirov, Dagestan's interior minster, said a collection of Kalashnikov rifles, machine guns and a sniper rifle were found where the fighters were hiding in the mountains.
"There are at least three non-Russians with Arab nationality amongst the dead militants, perhaps even four," he said.
Fighting continued overnight as security forces scoured the hills south of Makhachkala.
Analysts have suggested that attacks on security forces could intensify as economic problems and heavy-handed police tactics push many young men to join the separatist fighters.
"The authorities are very weak in Dagestan ... The rebels are often used as a cheap killing force," Yulia Latynina, a commentator with the independent newspaper Novaya Gazeta
"The situation is basically getting worse."
Predominantly Muslim Dagestan, with a population of more than 2.5 million, is in the North Caucasus between Chechnya and the Caspian Sea.
The violence in Dagestan has worsened as neighbouring Chechnya has become more stable under the control of Ramzan Kadyrov, the Moscow-backed president.