Abdullah made the remarks in a meeting with new Iranian Ambassador to Afghanistan Bahador Aminian in Kabul.
Aminian, for his part, said that Afghanistan is a close friend of Iran, and added that Iran regards the development and security of Afghanistan as its own security and development.
"Relations between the two countries are based on common values and historical-cultural ties," he said.
The two sides also exchanged views over presidential elections in Afghanistan and steps to reach peace as well as development of bilateral relations between Iran and Afghanistan.
Aminian had on December 14 submitted his credentials to Afghan President Ashraf Ghani.
Iran has always tried to help the Afghan government to establish peace in the country.
Iranian Foreign Minister Mohammad Javad Zarif and Afghan President Ashraf Ghani in a meeting in Kabul earlier this month reviewed peace talks among Afghan groups in the war-torn country.
Zarif and President Ghani met on the sidelines of the 8th Ministerial Conference of the Heart of Asia-Istanbul Process.
During the meeting in the Afghan capital, both sides discussed current political developments in Afghanistan.
The conference held in Istanbul is themed "Peace, Partnership, Prosperity" and is aimed at bolstering regional cooperation as well as dialogue towards Afghanistan.
The Heart of Asia-Istanbul Process discussed peace, regional participation, and cooperation for the future of Afghanistan.