Three others were wounded during the attack, said Mohamed Nur, a traffic police officer. One of the assailants was killed in a gun fight after the blast.
"It was a suicide car bomb that hit the gate. We also shot dead an armed militant who wanted to storm the building," Nur told Reuters.
Local residents said the front of the traffic headquarters was damaged extensively. The blast also destroyed nearby shops.
There was no immediate claim of responsibility.
The al Shabaab militant group, which wants to topple Somalia's government, carries out frequent attacks on military targets and civilian facilities like hotels and restaurants, mostly in the capital Mogadishu.