The Democratic US leader told the Republican president-elect his administration would “do everything we can to help you succeed, because if you succeed, then the country succeeds”.
Mr Obama said his talks with the billionaire political novice, held barely 36 hours after his upset election victory over the Democrat Hillary Clinton, were “wide-ranging”.
“We talked about foreign policy. We talked about domestic policy,” Obama said. “We talked about some of the organisational issues in setting up the White House.”
“This was a meeting that was going to last for maybe 10 or 15 minutes,” Mr Trump said.
“The meeting lasted for almost an hour and a half. And it could have, as far as I’m concerned, it could have gone on for a lot longer.”
“I have great respect,” he added, saying: “I very much look forward to dealing with the president in the future, including counsel.”
The two men ended the historic White House encounter with a handshake and refused to take questions.
The meeting had potential to be an awkward one – the two had traded barbs during the heated campaign for the White House, with Mr Obama describing the celebrity businessman as “uniquely unqualified” to be president.
Mr Trump, 70, championed the so-called “birther movement” challenging that Mr Obama was actually born in the United States – a suggestion laden with deep racial overtones – only dropping the position recently.
But in the day after Mr Trump’s shock election win, which virtually no poll had predicted, both sides spoke of healing the deep divisions sown in the bruising two-year battle for the presidency.
Ms Clinton, holding back the bitter disappointment of not becoming America’s first female president, urged the country to give Trump a chance.
“We owe him an open mind and the chance to lead,” she said Wednesday in a concession speech.
Mr Trump headed from the White House to Capitol Hill for a lunch meeting with House Speaker Paul Ryan. Vice President-elect Mike Pence was also to attend.
Mr Ryan, who had distanced himself from Mr Trump in the final month of the campaign, has pledged to “hit the ground running” and work with Mr Trump on conservative legislation.