On Twitter, he blasted "haters and fools", who, he said, do not encourage good relations between the countries.
Earlier he said Mr Putin told him he was insulted by allegations of Russian interference in the 2016 US election.
The US intelligence community has previously concluded that Russia tried to sway the poll in Mr Trump's favour.
"He said he absolutely did not meddle in our election," the US president said.
However, after intense criticism, Mr Trump clarified hat he supported US intelligence agencies in their conclusion. "As to whether or not I believe it or not, I'm with our agencies. I believe in our... intelligence agencies," he said.
"What he believes, he believes," he added, of Mr Putin's belief that Russia did not meddle.
The two leaders had no formal bilateral talks during the Asia-Pacific Economic Co-operation (Apec) event, but meet in passing on three occasions. They spoke about the Syria crisis and the election allegations, according to Mr Trump.
Republican Senator John McCain, a vehement critic of Mr Trump, called him naive for "taking the word of a KGB colonel over that of the American intelligence community".
A CIA statement passed to US media said: "The intelligence assessment with regard to Russian election meddling has not changed."
President Trump has refused to acknowledge a reported assessment by the CIA and other intelligence agencies that Russia was behind the hacking of the Democratic National Committee (DNC) in the run-up to last year's presidential election.
The contents of the emails, passed to Wikileaks and posted online, were embarrassing to the Democrats and shook up the presidential campaign, which ended in defeat for Hillary Clinton.
In addition to the Mueller inquiry, congressional committees have been set up to carry out their own investigations.
Relations between the US and Russia have been strained for years, with the Kremlin long accusing Washington of seeking to sway elections in Russia and other ex-Soviet states including Ukraine and Georgia.
While Russian hackers are widely suspected of involvement, there has been no conclusive link to the Kremlin.
Denying that Russia had tried to interfere last year by fostering contacts with Mr Trump's campaign, Mr Putin told reporters in Vietnam: "Everything about the so-called Russian dossier in the US is a manifestation of a continuing domestic political struggle."
He said he believed Mr Putin had been "very insulted by" the allegations and that was "not a good thing" for America.
"He [Putin] said he didn't meddle," he added. "I asked him again."
Asked if he believed Mr Putin, he replied, "He is very, very strong in the fact that he didn't do it. You have President Putin very strongly, vehemently says he has nothing to do with that. Now, you are not going to get into an argument, you are going to start talking about Syria and the Ukraine."
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