In Athens on a leg of his final world tour, Mr Obama vowed that his team would "do everything we can to support the smoothest transition possible" to the new presidency.
He said democracy "depends on peaceful transition of power especially when you don't get the results you want".
Mr Trump won last week's US election.
He will be sworn in as president in January 2017.
Looking back through achievements of his eight-year presidency, Mr Obama singled out the deal with Iran on nuclear weapons and Paris climate change agreement..
President-elect Trump had criticised both deals during his presidential campaign.
Mr Obama's last tour will also take in Germany and Peru.
US President Barack Obama has admitted he and Donald Trump "could not be more different" but said American democracy was "bigger than any one person".