Putin said on state television, according to the Associated Press, "Those who can be caught must be taken alive and brought to trial."
The AP also reported Ingushetia's acting interior minister, the health minister and a deputy interior minister were among the 47 law enforcement officials killed.
A spokesman for the Russian army blamed the attack on Chechen and Ingush rebels and "possibly" foreign fighters.
"The attacks were clearly saber rattling, aimed to demonstrate the rebels' effectiveness to attract funding from foreign terrorist networks," Maj. Gen. Ilya Shabalkin said, according to the AP.
Russian officials have accused some Chechen fighters of ties with the al-Qaida terrorist network and to Taliban training camps in Afghanistan.
Monday's attacks further undermined Russia's ability to defeat separatists in Chechnya who have often launched attacks in neighboring regions. In May, a bomb in the Chechen capital of Grozny killed the republic's President Akhmad Kadyrov, the man backed by Putin to bring peace to the region.
The Grozny bombing followed a series of suicide bombings in Russian cities that began when fighting resumed between Chechen rebels and Russian forces in 1999. At least 570 have died and many more have been injured since July 2000.
The attacks in Ingushetia began shortly before midnight Monday when about 100 rebels armed with grenades and rocket launchers entered the city of Nazran and seized the regional Interior Ministry, the AP reported. According to Russian Interior Minister Rashid Nurgaliyev, 15 security officers defended the building into the night until backup troops arrived in personnel carriers.
At least five militants were killed and others were captured in the coordinated raids in the city of Nazran and villages of Karabulak and Yandare. Others fled into the forests near the border of Chechnya and Ingushetia, the AP reported. A United Nations aid worker also died in the crossfire.
The region of Ingushetia, formerly part of the Chechen-Ingush Republic, gained separate status in 1991 after Chechnya declared independence. Chechnya's separatist President Aslan Maskhadov warned last week that insurgents could be planning new attacks.