The attackers opened fire on a vehicle as it drove in Garissa town, which lies some 120 kilometres (75 miles) from the border with war-torn Somalia, and where Kenyan troops are fighting Somali Islamist insurgents.
“It looks like the attackers were after the people in the vehicle, because they just opened fire at them and escaped,” regional police chief George Losku said.
A police officer was shot dead in the town on Monday night.
Attacks have increased in Kenya since Nairobi sent soldiers into Somalia in late 2011 to fight Al-Qaeda-linked Shabaab insurgents, who have vowed to take revenge on Kenya.
However, while the violence has been blamed on the insurgents or their Kenyan supporters, the Shabaab have not claimed responsibility for any of them.
Guns are common in the impoverished northeastern region, which hosts more than 500,000 Somali refugees.
Despite losing a string of key towns to African Union troops, allied Somali forces and Ethiopian soldiers, the Shabaab are still active in large areas of southern Somalia.