The letter revealed some seven months after the handover of power, revolves around one central bit of advice: beyond the bitterness and brutality of political combat and power struggles, never lose sight of the importance of democratic institutions.
Publication of the missive, just shy of 300 words, comes at a difficult and chaotic time in the Trump presidency that has seen intense criticism even from Trump's own Republican Party for his lack of clarity - and of moral leadership - after the racist violence in Charlottesville, Virginia.
"Dear Mr. President," the letter begins, going on to congratulate Trump for his "remarkable run" to the White House and to offer well wishes "as you embark on this great adventure."
The letter was obtained by CNN "from someone Trump showed it to," according to the channel.
"Millions have placed their hopes in you, and all of us, regardless of party, should hope for expanded prosperity and security during your tenure," it says.
President Barack Obama shake hands with President-elect Donald Trump during the Presidential Inauguration at the US Capitol in Washington, DC, in 2017. Source: AAP
Obama had pounded Trump during the presidential campaign with a rare virulence, saying the real estate mogul was, in his eyes, a danger to American democracy.
"The fate of the republic is in your hands," Obama told voters in North Carolina just days before the November 8 election.
But in the aftermath of Trump's shocking victory, with Democrats still badly shaken by the surprise defeat of Hillary Clinton, Obama insisted on the importance of a peaceful and constructive transition to the billionaire populist, even receiving him in the Oval Office in a meeting that would once have seemed unimaginable.
Emphasising that the US presidency is a "unique office" with no "clear blueprint for success," Obama nonetheless offers a few carefully worded suggestions to his successor.
Insisting on the importance of "indispensable" American leadership in the world, he emphasises the need for a president to act thoughtfully and responsibly.
"It's up to us, through action and example, to sustain the international order that's expanded steadily since the end of the Cold War, and upon which our own wealth and safety depend," Obama says.
'Good luck and Godspeed'
He also exhorted Trump not to let the vicissitudes of daily politics overshadow the long march of American democratic values.
"We are just temporary occupants of this office," he writes. "That makes us guardians of those democratic institutions and traditions - like rule of law, separation of powers, equal protection and civil liberties - that our forebears fought and bled for.
"Regardless of the push and pull of daily politics, it's up to us to leave those instruments of our democracy at least as strong as we found them."
The final bit of advice extended by the 44th US president to the 45th took a more personal tone: "Take time, in the rush of events and responsibilities, for friends and family. They'll get you through the inevitable rough patches."
Saying that he and his wife Michelle "stand ready to help in any ways which we can," Obama closes with a wish of "good luck and Godspeed," signing the letter, "BO."
Trump said publicly that he was touched by the letter, though he has not seen Obama since the handover of power.
Of all the letters from outgoing to incoming presidents, the one left on January 20, 1993, by Republican George H.W. Bush to Democrat Bill Clinton stands out for its dignity and class.
Bush wrote to his successor - the man who had deprived him of a second term in office - that "your success now is our country's success."
Evoking the "sense of wonder and respect" for the office that he felt when he became president in 1989, Bush said, "There will be very tough times," but then added: "Just don't let the critics discourage you or push you off course."
Bush concluded: "I am rooting hard for you. Good luck."
