Mr Yameen secured 51.3% of the vote, compared with 48.6% for ex-President Mohamed Nasheed, the Election Commission said.
Mr Nasheed had won 47% in the first round this month, just short of the 50% needed for outright victory, in an election process mired in controversy.
He has accepted defeat, saying he supported the democratic process.
He had been seeking to regain power after he was forced to resign in 2012.
Mr Yameen, the half-brother of Maumoon Abdul Gayoom - who ruled the nation for 30 years - will be sworn in on Sunday.
Mr Nasheed, of the Maldivian Democratic Party (MDP), said: "I graciously accept defeat. We lost by a very small margin. Democracy is a process. It is up to us to make it work."
He added: "The MDP has always asked for a government elected by the people. Today is a happy day for the Maldives - we now have an elected government."
Mr Yameen said: "It's now time to bring peace, the people have decided. It's now time for development."
He said his priority was the economy, notably stabilising the rupiah and creating jobs.
There have been months of legal and political wrangling over the election, causing concern among foreign diplomats.
The Supreme Court annulled a first vote on 7 September amid allegations of electoral fraud.
A planned re-run on 19 October was halted and the election finally took place on 9 November.
However, a first attempt to hold a run-off the following day was cancelled hours before voting was due to begin.
Police prevented election officials from distributing voting materials, citing a court order requiring all candidates to approve the voter register.
The European Union had said it would take "appropriate measures" if the election was not run on Saturday.
Abdulla Yameen has won the presidential election run-off vote in the Maldives, officials say.