The Kremlin leader also reminded the gathered media that he used to be an intelligence agent, adding: "I do know how dossiers are made up."
Both men's unity was apparent throughout the meeting, with the leaders suggesting Moscow and Washington could jointly conduct criminal investigations into Russian intelligence officials accused of hacking during the election campaign.
Asked if Russia could extradite 12 Russian military intelligence officers indicted in the US last week on charges of hacking into the Democratic election campaign, Putin challenged the US to take advantage of a 1999 agreement envisaging mutual legal assistance.
He said the agreement would allow US officials to request that Russian authorities interrogate the 12 suspects, adding that US officials could request to be present in such interrogations.
When asked to condemn Russian meddling in the election, Trump instead complained about a Democratic National Committee computer server and emails belonging to Hillary Clinton, the Democrat he defeated to win the presidency.
The pair were also as one on the issue of Syria - both agreeing a resolution was needed for the humanitarian crisis.
And as for the relations between the US and Russia, which Trump described as being in the worse state it had ever been before the summit, there was "no reason" for the difficulties and tension, Putin said.
The Kremlin leader said he and Trump have exchanged "concrete suggestions on this topic" and will continue "work along the entire nuclear disarmament dossier".
In the most bizarre moment of the press conference, Putin took an official Russia FIFA World Cup 2018 football and symbolically handed it to Trump.
The US hosts the tournament in 2026, four years after Qatar, and the Russian leader said the ball was now in the President's court.
Trump said he hoped his country could do as good a job as Russia, before tossing the ball from the podium to his wife Melania sitting in the front row, saying the ball would be going straight to his football-mad 12-year-old son Barron.
Before the press conference had even started, a man - thought to be a journalist from the US magazine The National - was forcibly ejected from the room.
He was holding up a sign which read: "Nuclear weapon ban treaty".