The scale of the democracy campaigner's apparent victory is close to her party's performance in the 1990 constituent assembly elections, which was ignored by Burma's military rulers.
Ms Suu Kyi and other leading NLD figures spent much of the last two decades under arrest, but she was freed in 2010 in a surprise move by the regime.
Yesterday marked Ms Suu Kyi's first election contest in the poor Kawhmu township just outside Rangoon, which her party said she won with 99pc of the vote.
Hundreds of NLD workers began to gather on the road outside the party's ramshackle headquarters in Rangoon to celebrate their anticipated victory.
The numbers swelled to thousands as officials led supporters wearing their red party colours and waving peacock flags. They became increasingly ecstatic as results were broadcast on a giant television screen outside.
Sonny Nyunt Thein, the head of the influential Myanmar Egress policy centre, which has advised President Thein Sein on several of his democratic reforms, said early results indicated that the NLD would win at least 75pc of the 45 by-elections in a "landslide victory". The NLD claimed last night they were on course to win 40 seats.
If confirmed, the results would give the NLD a bloc to create a vocal opposition in the parliament and put it in pole position to win the next general election in 2015.
NLD sources said the party had won all 11 seats where counting was completed, including one in President Thein Sein's new capital, Naypyidaw, where many residents work for his government. "It looks like a landslide, more than 75pc and that means a good image for the country," said Mr Nyunt Thein.
Official results are expected within a week. The scale of her apparent victory would counter claims, including some from Ms Suu Kyi, that the election had not been free and fair -- one of the key conditions for the lifting of United States and European Union sanctions.
"If (her victory) is like this, it's clearly a free and fair election. If the international community accepts this election as clean, I'm sure they will lift the sanctions," Mr Nyunt Thein added.
The voting was marred by fresh claims that several villages were excluded from lists and that ballot papers in Ms Suu Kyi's constituency were marked with wax so NLD votes could be wiped away.
EU observers had previously warned that they would not be able to declare the elections free and fair because they had not been invited in sufficient time to carry out a full and credible inspection mission. (© Daily Telegraph, London)