U.N. Secretary-General Ban Ki-moon also commended the country "for the peaceful and largely orderly manner" in which the elections were held, according to his spokesperson.
The NLD prediction was based on the party's own estimates, according to party member Thae Da Win Aung. It was still unclear whether the NLD had won the 44th seat, she said.
Suu Kyi, 66, led her party to a landslide victory the last time Myanmar held multiparty elections, in 1990. But the junta ignored the results and placed her under house arrest.
Released in November 2010, Suu Kyi was allowed to crisscross the country to rally support for the NLD for Sunday's race.
The United States announced in January that it would exchange ambassadors with Myanmar after the regime released political prisoners.
Clinton visited Myanmar in December -- a historic trip marking the first time a secretary of state had been to the country in more than 50 years -- and British Foreign Secretary William Hague visited the following month.
On Sunday, Clinton said she had been impressed with her visit.
"I was very touched by the visit that I made and the commitments I received from members of the government who were quite sincere in their desire to move their country forward," she said, even as she underlined that the United States wanted to see "continuing progress."
The NLD fielded a candidate for every one of the 45 seats up for grabs. But the election commission rejected one candidate, apparently because his parents had foreign residency. The NLD has said it plans to challenge his exclusion.
The government had promised the vote would be free and fair, and allowed international observers to monitor the polling.