Magomedtagirov, who has run the interior ministry since 1998, was not in the car when it exploded and television footage showed his wrecked Toyota Landcruiser.
"It seems obvious to me that it was an assassination attempt on the minister because the explosion occurred when his car was driving by," Anzhela Martirosova, an interior ministry spokeswoman, told Russian television.
The attack comes one day after Russian military forces killed at least four separatists during a special operation in Ingushetia.
Police said a suspected associate of the four alleged fighters killed himself on Sunday by detonating a hand grenade.
Saturday's bomb attack in the Dagestani capital marks the second suspected attempt on Magomedtagirov's life in less than a year.
In August last year, his convoy was hit by an explosion and gunfire on the way to the site of a car bombing that killed a prosecutor.
Two police officers died in that attack, but Magomedtagirov was unhurt.
NTV television reported that the police officer who was shot was the son of a friend of Magomedtagirov.
Dagestan, an ethnically mixed mostly Muslim region, has long been affected by violence caused by criminals and fighters.
Violence has been linked to the conflict in Chechnya, political infighting, and turf wars over lucrative economic activities.
The regional interior ministry says five police officers have been killed in Dagestan this year, some of them in confrontations with suspected fighters or criminals.