The country has been dominated by the military for half a century through direct junta rule and since 2011 by a quasi-civilian government run by its allies.
But Suu Kyis National League for Democracy (NLD) is on the verge of tipping the balance of power after capturing more than 85 percent of seats declared so far from Sundays election -- a huge stride in the partys long democracy struggle.
Obama "called President Thein Sein this morning to congratulate him and the government for successfully holding a historic free and fair general election," Myanmars Information Minister Ye Htut said on his official Facebook page, AFP qouted.
The American leader has thrown his weight behind Myanmars reform process and its pro-democracy figurehead Suu Kyi, visiting the country twice since the end of outright military rule in 2011.
He has urged the country to tackle religious intolerance and promote full democracy. He has also highlighted the plight of the ethnic Rohingya Muslims, tens of thousands of whom were excluded from voting.
In his call, Obama told his counterpart to be "proud of.. the milestone election" and praised "the presidents brave reforms", said Ye Htut, a close Thein Sein aide.
There was no immediate confirmation of the call by the US State Department.
By Thursday morning the NLD had swept up 273 seats, 56 short of an outright majority. It is almost certain to smash through that marker, with more official results due to be released on Thursday.
In statements released via Facebook, Thein Sein and the powerful army chief Min Aung Hlaing congratulated Suu Kyis party, vowed to respect the poll result and work with the new government.
The army chief repeated his position to military top brass, vowing "co-operation with the new government during the post-election period," according to a statement posted on his Facebook page Thursday.
Suu Kyi on Wednesday called for national reconciliation talks with the army chief and Thein Sein, stressing the need for a peaceful transition.
Many NLD supporters remain deeply suspicious of the army and its
parliamentary allies, who are notorious for political sleight of hand and crackdowns on democracy movements that have left hundreds dead and thousands jailed.
Suu Kyis party won a 1990 election by a landslide only for the army to ignore the result and tighten its grip on power.
In an indication of the days and weeks of manoeuvring through Myanmars treacly politics that lie ahead, NLD spokesman Nyan Win said the call for talks was "our first move... to help a smooth transfer of power."