A car bomb was detonated at about 0830 (0430 GMT) outside the offices of the local interior ministry and the FSB security agency in the town of Kizlyar.
The second bomb went off about 35 minutes later on the same street.
Russia is on alert after double suicide bombings on the Moscow Metro on Monday morning, which killed 39 people.
Prime Minister Vladimir Putin has called on security forces to "scrape from the sewers" those responsible. Investigators say they believe the bombers were linked to militants operating in the North Caucasus.
Last month, Chechen rebel leader Doku Umarov threatened that "the zone of military operations will be extended to the territory of Russia... the war is coming to their cities".
Violence in Dagestan has escalated following a crackdown on militants in Chechnya. In June, its interior minister was shot dead.
Officials said the chief of police in Kizlyar, Col Vitaly Vedernikov, was killed in the second of Wednesday's blasts.
He was in a crowd gathered at the site of the car bomb, which a suicide bomber infiltrated before blowing himself up, they added.
Kizlyar is situated close to Dagestan's border with Chechnya.