In his victory speech, Soli described the win as a "moment of happiness, hope and history".
"But it's been a journey that has ended in the ballot box, because the people willed it".
"The message is loud and clear". "The Maldivian people have decided what they want". I have accepted the results from yesterday.
The strongman president of the Maldives on Monday conceded defeat in elections, easing fears of a fresh political crisis in the archipelago at the centre of a battle for influence between India and China.
Former President Mohamed Nasheed congratulated President-elect Solih for his hard work and for keeping the opposition front together and united. A statement from spokeswoman Heather Nauert noted the reported opposition victory and urged "calm and respect for the will of the people" as the election process was being concluded.
A spokesman for Maldives' Election Commission said official results would not be announced until September 29, allowing a week for parties to challenge the results in court.
China has spent hundreds of millions of dollars on infrastructure projects in the Maldives, which critics, including the political opposition, warn amount to "debt-trap diplomacy" that weighs down the recipient country with loans in order to secure a naval base as repayment.
During the election campaign, Yameen had presented himself as a Maldivian nationalist focused on economic development, pointing to infrastructure projects built during his term, including a 2km bridge linking Male to the worldwide airport that opened earlier this month.
A crackdown saw former president Maumoon Abdul Gayoom - Yameen's half-brother - jailed along with the Chief Justice and another Supreme Court justice amid accusations of an imminent coup.
Voters debated the relative merits of the two candidates in front of a polling station at the Imauddin School.
One of the most senior politicians in the Maldives, and has for years been calling for democratic reform.
While presidential terms usually end on November 11, President Yameen's term will end on November 17, the same day Solih and his running mate Faisal Naseem will take their oath of office.
The European Union and USA have voiced concerns about the election and have threatened to impose targeted sanctions if the democratic situation does not improve. But the run-up to the election was not an easy task for Mr. Solih.
Some 262,000 people in the archipelago - famed for its white beaches and blue lagoons - were eligible to vote in an election from which independent global monitors have been barred. "But we did not get the required visa", he told Reuters.
The election commission released provisional results earlier Monday showing Solih had won the South Asian island nation's third-ever multiparty presidential election with 58.3 percent of the vote. "We call on all stakeholders to maintain an environment conducive for a peaceful transfer of power". Waheed, who believes Yameen has weekend democracy in the Maldives, said a free and fair election "will change this regime".
The European Union said Friday that it was not sending election observers because the Maldives had failed to meet the basic conditions for monitoring.