In the first attack, a suicide bomber rammed an explosives-laden car into a spy agency facility in the eastern town of Jalalabad, 150 kilometres (90 miles) east of Kabul.
It was followed by a similar attack on a police base in Logar, 70 kilometres south of the capital, officials said.
“There was a suicide car bombing in the intelligence facility in city district 2. Two intelligence workers were martyred and three others were wounded,” Jalalabad police spokesman Hazrat Hussain Mashriqiwal told AFP.
He said the attacker rammed his vehicle into the gate of the walled compound and detonated his bombs.
Police in Logar said the attack there hit a police base along the highway leading to Kabul and killed one police officer.
“A suicide bomber driving a van hit the gate of the police post near Pol-i-Alam. One police was killed and two other police were wounded,” Logar provincial police chief Abdul Saboor Nasrati told AFP.
Taliban spokesman Zabihullah Mujahed claimed responsibility for the Logar bombing. There was no immediate claim for the Jalalabad attack.
The Taliban are the main insurgent group behind suicide bombings in Afghanistan in a long-running insurgency aimed at toppling the Western-backed government in Kabul.