In a televised forum on Wednesday, Mr Trump had praised Mr Putin`s "great control" and 82% approval rating.
Mr Trump and rival Hillary Clinton had taken questions from military veterans.
Mr Obama said: "I don`t think the guy`s qualified to be president of the United States and every time he speaks, that opinion is confirmed."
The president pointed to the diplomatic work he had faced at both the Asean summit in Laos and the earlier G20 meeting in China.
He said: "I can tell you from the interactions I have had over the last eight or nine days with foreign leaders that this is serious business.
"You actually have to know what you are talking about and you actually have to have done your homework. When you speak, it should actually reflect thought-out-policy you can implement."
Mrs Clinton, meanwhile, pilloried Mr Trump for having suggested US military leaders had been "reduced to rubble", accusing him of having "trash-talked American generals".
In a rare press conference, she said on Thursday morning: "That`s how he talks about distinguished men and women who`ve spent their lives serving our country, sacrificing for us."
Mr Trump had told the forum in New York that the Russian president had "been a leader far more than our president has been".
Quizzed by NBC host Matt Lauer on his previous complimentary remarks about Mr Putin, Mr Trump responded: "He does have an 82% approval rating."
"I think when he calls me `brilliant`, I`ll take the compliment, OK?" said the businessman, adding that Mr Putin had "great control over his country".
Mr Trump also said that, as a result of the confidential intelligence briefings he has been entitled to as an election candidate, he had been "shocked" at how the president, Mrs Clinton and current Secretary of State John Kerry had done "exactly the opposite" of what intelligence experts had told them.
`Unilluminating blather` - US media`s initial response
Gabriel Debenedetti writes on the Politico website that neither candidate did much to advance their cause, with Hillary Clinton spending "a third of the time fending off questions about her emails" while "Donald Trump struggled to explain his secret plan to defeat the Islamic State."
Time agreed, citing "plenty of unilluminating blather spewed by both candidates". It said: "The most dispiriting thing was the grim view of the world the candidates gave Americans, with their relentless focus on fighting and terror... There was scant optimism."
One focus was on the performance of NBC moderator Matt Lauer. The New York Times pronounced that the "consensus afterwards was not kind". "Mr Lauer found himself besieged... by critics of all political stripes, who accused the anchor of unfairness, sloppiness and even sexism in his handling of the event."
Lauer`s main miss, it seemed, was not to press Mr Trump when he said he had not supported the war in Iraq. Vox was among those pointing to a 2002 interview with radio host Howard Stern that contradicts this.
In the forum, Mr Trump also said: "I was totally against the war in Iraq."
This appeared to contradict a statement in a 2002 interview with radio host Howard Stern and the forum`s moderator, Matt Lauer, came in for intense criticism after the event for not pressing Mr Trump on the statement.
Mr Obama said in Laos: "The most important thing for the public and the press is to just listen to what he says and follow up and ask questions to what appear to be either contradictory or uninformed or outright wacky ideas".
Mrs Clinton had found herself once again on the defensive during the forum over her private email server.
The forum offered a preview of the questions they will face in their three forthcoming presidential debates, the first at Hofstra University near New York on 26 September.
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