Obama looks for a smooth transition of power
President Obama said it was "no secret" that he and Mr Trump have had some "pretty significant differences" when speaking about the result for the first time in public.
However, he said he also had significant differences with George Bush when he took over at the White House eight years ago, and that "one thing you realise quickly in this job is that the presidency and vice presidency is bigger than all of us".
Speaking in Washington on Wednesday, Mr Obama said what America needs now is "a sense of unity" and that "ultimately, we are all on the same team".
Urging fellow Democrats to put aside their disappointment after the devastating defeat, he kept his remarks focused on ensuring a successful transition for the billionaire President-elect.
He also said he had been "heartened" by Mr Trump's call for unity following the announcement of the result.
Mr Obama said he was instructing his team to make sure there is a peaceful transfer of power to the man who will become the 45th president of the United States, after surpassing the 270 Electoral College votes necessary to win.
Paying tribute to Hillary Clinton, he said he "could not be prouder" of the Democrat for her "extraordinary life of public service".
He continued: "She was a great first lady, she was an outstanding senator for the state of New York and she could not have been a better secretary of state."
Mr Obama said her candidacy and nomination sent a message to daughters all across the country that "they can achieve at the highest levels of politics", and that he was confident that she would, along with husband Bill Clinton, "continue to do great work for people around the world".
His speech came after Mrs Clinton addressed tearful backers in New York following her defeat, appealing to supporters to accept the election result and give Mr Trump the "chance to lead".