Pakistan has congratulated the people of Maldives on the peaceful conclusion of its Presidential election.
Yameen, who was widely tipped to retain power, had jailed or forced into exile nearly all of his main rivals.
There had been concerns that Yameen, who has borrowed hundreds of millions of dollars from China for an infrastructure blitz, might not accept the outcome.
"I would like to call upon President Abdulla Yameen and ask him to respect the will of the people and immediately begin the smooth transition of power", Solih said at the news conference at midnight.
The Indian Ministry of External Affairs released a statement commenting on the unexpected majority of 58.4 percent achieved by candidate Solih, and urged the Elections Commission (EC) to publicise a formal result at the earliest opportunity.
"This is a moment of happiness, a moment of hope", Solih told reporters in the capital Male.
Incumbent Abdulla Yameen, who had cultivated ties with both Beijing and Saudi Arabia, conceded defeat after the Election Commission said Solih had won Sunday's election by a margin of 16.7 percent. The reason why many raised doubts whether if the elections would be free and fair.
Ahamed Fiasal, a 39-year-old IT business owner who voted for Solih, said the election result was surprising because "no one thought that Yameen would lose like this".
"The will of the people has spoken", said Solih.
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Independent worldwide monitors were barred from the election and only a handful of foreign media were allowed in to cover the poll.
A police raid on Solih's main campaign office the night before the election was seen as a worrying sign that Yameen would attempt to "muzzle his way" to re-election, according to Hamid Abdul Gafoor, an opposition spokesman and former Maldives lawmaker now based in Colombo, Sri Lanka.
"For many of us this has been a hard journey, a journey that has led to prison cells or exile".
For the first time in three years, it was permitted this month to hold a rally, after the government came under pressure for not issuing permits in the past.
Yameen's supporters were also vocal.
In February, Yameen declared a state of emergency, suspended the constitution and ordered troops to storm the Supreme Court and arrest judges after they had ordered the release and retrial of those jailed after politically motivated trials.
Foreign monitors said Yameen failed to carry out any large-scale fraud thanks to intense global and local scrutiny from civil society groups.
Mr. Yameen has been racked by accusations of corruption, including reports that he plans to sell some of the 1,200 islands that make up the Republic of Maldives for his personal gain, a charge he has denied. Others said they could not get visas in time.