Putin has built much of his reputation on harsh statements, but he did not elaborate on what kind of retribution he had planned or against whom, in his comments Tuesday carried by Russian news agencies.
No claims of responsibility have been made for Monday's explosion at Domodedovo Airport which also left 180 people injured.
Suspicion is likely to fall, however, on Islamist separatist insurgents from Chechnya or elsewhere in Russia's restive Caucasus region who have been battling Russian authority for over 15 years.
Chechen insurgents have claimed responsibility for an array of attacks in Moscow and elsewhere in Russia, including a double suicide bombing on the capital's subway system in March 2010 that killed 40 people. They also have hit Domodedovo Airport before, with two suicide bombers slipping through its security in 2004 to kill 90 people.
Medvedev described security at the airport as being in "a state of anarchy" and said management there must bear key responsibility for security failures that contributed to Monday's blast. He also said government security officials would be held accountable for any lapses found.
Airport management objected, saying transport police were responsible for the inspection of people coming into the international arrivals area, where the bombing took place.
The finger-pointing could undermine confidence in Russia's security ahead of Medvedev's high-profile appearance this week trying to attract investors at the World Economic Forum in Davos, Switzerland. Monday's attack also called into question Russia's ability to safely host major international events like the 2014 Winter Olympics in Sochi and the 2018 World Cup.
Putin rose to power largely on his tough-against-terror image, including a famous vow that Chechen rebels would be hunted down and killed "in the outhouse." But despite launching the second Russia-Chechnya war and pushing harsh against suspected rebels, he was unable to wipe out the Chechen insurgency during his 2000-2008 presidency.
In addition, it's unclear what levers Putin could push now if he aims to exact retribution. After the Beslan school hostage crisis in 2004 that killed more than 330 people, half of them children, Putin pushed through changes to make regional governors appointed rather than elected. Still, further attempts at consolidating Russian government control could provoke a backlash from an opposition movement that has grown in recent months.
Aviation security experts have been warning since the Sept. 11, 2001, attacks that the crowds at many airports present tempting targets to suicide bombers.
Monday's bombing exposed the unprotected underbelly of airport security - the international arrivals area, packed with families, taxi drivers and businesspeople, all of whom do not have to go through airport security. Few airports in the world control the entrances to such areas.
CBS News correspondent Mark Phillips reports that a closed circuit security camera captured the moment when a typical airport scene - passengers moving through the international arrivals area of Domodedovo - was turned into a scene of carnage.
At first people seem stunned, but as the smoke settled, the extent of the bomb's effect became apparent.
"That moment I saw people die on the floor and people lying on luggage trolleys and it was coming to me... it was terrible," recounted Danish traveler Frank Osterdijh.
The Emergencies Ministry said the dead included one person each from Britain, Germany, Austria, Ukraine, Tajikistan. Kyrgyzstan and Uzbekistan; 16 were Russians and the remaining 12 had not been identified. A further 110 people, including nine foreigners, were hospitalized.
Medvedev postponed his departure for Davos, where he is to be the main speaker at the opening session Wednesday. The Kremlin said he still planned to go.
Russia's attractiveness for investors had already been shaken in December when ex-oil tycoon Mikhail Khodorkovsky was sentenced to six more years in prison. Khodorkovsky has been jailed since 2003 in a case critics say is revenge by Putin for his support of opposition politicians.
The Russian president, often seen as submissive to Putin, appeared to be trying to assert his power Tuesday by suggesting that officials at both the Interior Ministry and the Federal Security Service could be at fault.
"I instruct the Interior Minister to suggest which ministry officials responsible for transport security could be dismissed or face other sanctions," he said, and made similar instructions to the security service.
He also called for "total examination" of passengers and baggage at key transport centers. "This will make it longer for passengers, but it's the only way," he said.
It was the second time in seven years that Domodedovo was involved in a terrorist attack: In 2004, two female suicide bombers penetrated the lax security there, illegally bought tickets from airport personnel and boarded planes that exploded in flight and killed 90 people.
Conflicting reports emerged Tuesday about how the bombing was carried out. Some accounts, citing unnamed sources, said there were two bombers, one of them a woman.
Investigative Committee spokesman Vladimir Markin said Monday the attack was most likely by a suicide bomber and "attempts were being made to identify him."
Medvedev lashed out at Domodedovo management.
"What happened shows that obviously there were violations in guaranteeing security. And it should be answered for by those who make decisions there and by the management of the airport," he said.
The blast came at 4:32 p.m. Monday, when thousands of passengers and workers were in the terminal. They were sprayed with shrapnel containing screws and ball bearings, intended to cause as many casualties as possible, and thick smoke engulfed the terminal.
"There was lots of blood, severed legs flying around," said Yelena Zatserkovnaya, a Lufthansa official, who said airport workers used baggage carts to wheel the wounded to ambulances.
Built in 1964, Domodedovo is located 26 miles southeast of Moscow and is the largest of the three major airports that serve the capital, handling more than 22 million people last year.