Mr Obama spoke via video, as the coronavirus pandemic prevented the students from holding their graduation ceremonies in person.
Both speeches were filled with the sort of general life advice typical of such events, but they also contained several surprisingly political lines – including some barely disguised swipes at Mr Trump and his administration.
“More than anything, this pandemic has fully, finally torn back the curtain on the idea that so many of the folks in charge know what they’re doing,” Mr Obama said.
“A lot of them aren’t even pretending to be in charge.
“Doing what feels good, what’s convenient, what’s easy. That’s what little kids do.
“Unfortunately a lot of so-called grown-ups, including some with fancy titles and important jobs, still think that way. Which is why things are so screwed up.
“All those adults you used to think were in charge and knew what they were doing? Turns out, they don’t have all the answers. A lot of them aren’t even asking the right questions. So if the world’s going to get better, it’s going to be up to you.”
He never mentioned Mr Trump by name, but there was no mistaking the target of his words.
More broadly, Mr Obama argued the pandemic had “laid bare” the most “deep-seated” problems in American society.
In one of the speeches, directed specifically at graduates of historically black universities, he noted that African-Americans had been disproportionately affected by the virus.
“This pandemic has shaken up the status quo and laid bare a lot of our country’s deep-seated problems, from massive economic inequality to ongoing racial disparities, to a lack of basic healthcare for people who need it,” he said.
“It’s woken a lot of young people up to the fact that the old ways of doing things just don’t work. That it doesn’t matter how much money you make if everyone around you is hungry and sick. And that our society and democracy only work when we think not just about ourselves, but about each other.”
In a phone call with former members of his administration, Mr Obama labelled the US government’s response to the pandemic a “chaotic disaster”.
“What we’re fighting against is these long term trends in which being selfish, being tribal, being divided and seeing others as the enemy – that has become a stronger impulse in American life,” he said.
“It’s part of the reason why the response to this global crisis has been so anaemic and spotty.
“It would have been bad even with the best of governments. It has been an absolute chaotic disaster, when that mindset – of ‘what’s in it for me’ and ‘to heck with everybody else’ – when that mindset is operationalised in our government.”
Mr Obama’s comments were part of a broader pitch to his former staff and officials, urging them to rally behind the Democratic Party’s presidential nominee, Joe Biden.
They infuriated Mr Trump and his allies in the Republican Party, who said Mr Obama had breached a longstanding informal tradition that former presidents refrain from publicly criticising the current occupant of the White House.
The party’s most senior member of Congress, Senate leader Mitch McConnell, issued a particularly scathing response to Mr Obama during an interview with Mr Trump’s daughter-in-law Lara Trump, broadcast live by the President’s re-election campaign.
“I think President Obama should have kept his mouth shut,” Mr McConnell said.
“You know, generally, former presidents just don’t do that.
“I remember President George W. Bush and his father went right through eight years of Democratic administrations after they left office and kept their mouths shut, because they didn’t feel it was appropriate for former presidents to critique even the president of another party.
“I think it’s a bit classless, frankly, to critique an administration that comes after you.
“You had your shot. You were there for eight years. I think the tradition that the Bushes set up, of not critiquing the president who comes after you, is a good tradition.”
The President himself has spent the last 10 days hyping up “Obamagate”, which is the latest iteration of his theory that Mr Obama and officials in his administration spent their final weeks in office conspiring to sabotage him.
Mr Trump has repeatedly called it “the biggest political crime in US history”.
“What crime exactly are you accusing President Obama of committing, and do you believe the Justice Department should prosecute him?” a reporter asked.
“Obamagate. It’s been going on for a long time. It’s been going on from before I even got elected, and it’s a disgrace that it happened,” Mr Trump replied.
“And if you look at what’s gone on, if you look at now all of this information that’s being released – and from what I understand, that’s only the beginning – some terrible things happened, and it should never be allowed to happen in our country again.
“You’ll be seeing what’s going on over the coming weeks, and I wish you’d write honestly about it, but unfortunately you choose not to do so.”
“What is the crime exactly, that you’re accusing him of?” the reporter pressed.
“You know what the crime is. The crime is very obvious to everybody. All you’ve got to do is read the newspapers, except yours.”
Of course, the animosity between Mr Obama and Mr Trump has been simmering for far longer than the last fortnight. It goes back the better part of a decade now.
Mr Obama infamously mocked Mr Trump during the White House correspondents’ dinner in 2011. Before that, Mr Trump was one of the most prominent and persistent boosters of the racist conspiracy theory that Mr Obama was actually born overseas, and was therefore ineligible to be president.