Thousands of Romanians living abroad were allegedly unable to vote in the November 2 first-round presidential election due to an insufficient number of open polling stations in countries including France, Germany and Britain.
Prime Minister Victor Ponta finished first in the vote with 40 percent of the ballots, while his conservative rival Klaus Iohannis took 30 percent.
However, Iohannis won 46 percent of the vote among Romanians living abroad, while Ponta had 18 percent.
The second round is set for November 16.
Nearly 7,000 people gathered in Cluj in the northwest while another 1,500 protested in the western city of Timisoara, where the anti-communist uprising that led to the fall of dictator Nicolae Ceausescu began in 1989.
Protesters in Timisoara called for Ponta to resign, while hundreds of people also demonstrated in Bucharest, Oradea and Constanra.
“We came out in the streets to see to it that what happened last Sunday does not happen again on November 16,” said one of the protesters in Bucharest, Alexandru Alexe.
Outgoing President Traian Basescu on Saturday called on Foreign Minister Titus Corlatean to resign, while the minister has pledged improved organisation for the second round vote.
The vote is seen as a crucial test for the former communist country at a time when democracy has suffered setbacks in some neighbouring states such as Hungary, and as the Ukraine crisis has shaken relations between the European Union and Russia.
Whoever takes over the presidency will face pressing issues including recession and persistent accusations of corruption and bad governance.