Suu Kyi's NLD Party is winning 44 out of the 46 seats being contested in the by-elections but Suu Kyi's party will not become a threat to the majority or ruling party whose members came from the military junta which ruled the country for decades.
Supporters of Suu Kyi's NLD Party have converged and danced in the streets to cheer Suu Kyi and her party as they celebrate their victory.
"We have been waiting for this day for a long time. I'm so happy," said NLD supporter Kalyar, who goes by one name.
Reports of Suu Kyi's party victory came from sources at the NLD and no official confirmation yet coming from election officials. The official result of the election winners is expected to be announced within the week.
There were earlier reports of minor election irregularities but it is not yet known if these election offenses were widespread.
Several election observers
were invited by the government to observe the conduct of the by-elections which include representatives from the US, Asean and several European countries.
A clean and credible elections will be crucial to the eventual restoration of the country's diplomatic relations with several western democratic countries.
A free and credible elections could likewise lead to the lifting of economic sanctions imposed on the country by the US and other European countries.
The military-backed civilian government which took power after the March 2010 elections, has surprised many western countries with its reformist actions which include the relaxation on media, labor reforms, release of political prisoners and liberalized rules on freedom of assembly.
Suu Kyi swept her NLD party to victory in an election held two decades ago but the military regime did not allow her to take her post as the nation's Prime Minister. Instead, she was put on house arrest for most of the past 20 years since winning the election by landslide.
is a recipient of a Nobel Peace Prize for her democratic crusade in her impoverished and military-ruled country.