Relations between Riyadh and Tehran resumed earlier this year with the help of China.
Prince Mohammed and Mr Amirabdollahian discussed the latest international and regional developments, Saudi Arabia’s state news agency SPA said.
They also “reviewed relations between the kingdom and Iran, future opportunities for co-operation between the two countries and ways to develop them”, it said, apart from discussing regional and international developments.
"This morning, we had a 90-minute meeting with the Crown Prince of Saudi Arabia, Emir Mohammad bin Salman, in Jeddah," said Mr Amirabdollahian.
He called the meeting "a frank, beneficial and productive conversation based on neighbourhood policy".
"With the will of the heads of the two countries, stable bilateral relations were emphasised in all fields," Mr Amirabdollahian said. "We agree on security and development for all in the region."
Iran's state-owned broadcaster Press TV said Prince Mohammed, Mr Amirabdollahian and Saudi Foreign Minister Prince Faisal bin Farhan were meeting behind closed doors.
The meeting "is a great sign of ties between Iran and Saudi Arabia warming and expanding", its correspondent Gisoo Misha Ahmadi said from Jeddah
Mr Amirabdollahian arrived in the kingdom on Thursday and held talks with Prince Faisal.
“Relations between Tehran and Saudi Arabia are on the right track and we are witnessing progress,” Mr Amirabdollahian said during a news conference with Prince Faisal, who said the talks were “successful”.
Prince Faisal said the kingdom was hoping President Ebrahim Raisi of Iran would visit Saudi Arabia after an invitation from King Salman.
“Our meeting today is a continuation of the steps taken towards implementing the agreement to resume diplomatic relations, which represents a pivotal platform in the history of the two countries and the path of regional security," Prince Faisal said.
The trip comes as the two states are reopening diplomatic missions in each others' countries. Mr Amirabdollahian was accompanied by Alireza Enayati, Iran's new ambassador to the kingdom.