From the outset of the blockbuster encounter, the US property tycoon fired the key question that has weighed on his presidency, pressing the ex-KGB agent "on more than one occasion" on the vote interference claims, said Secretary of State Rex Tillerson, who was in the meeting.
But Tillerson's counterpart Sergei Lavrov quickly added that Trump "accepted" Putin's denial of Moscow's role in propelling him to the White House, after hearing the Russian out.
Over the course of two hours and 15 minutes of talks that covered not only the damaging claims but also a slew of global crises including the Syrian war, the two leaders clearly engaged, said Tillerson.
"Let me characterise: the meeting was very constructive, the two leaders I would say, connected very quickly," said Tillerson, adding "there was a very clear positive chemistry between the two".
"There are so many issues on the table... Just about everything got touched upon... Neither one of them wanted to stop," he said.
"I believe they even sent in the (US) First Lady at one point to see if she could get us out of there, but that didn't work either... We did another hour. Clearly she failed!"
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Growing rift
On the presidential election campaign trail last year, Trump said he hoped relations with Putin could be rebuilt after Russia's acrimonious ties with his predecessor Barack Obama.
But the billionaire had faced increasing pressure to take a tough line on Russia after allegations it interfered in the vote.
In the talks on the sidelines of the G20 summit, Trump pressed Putin on the issue, "and then... felt like at this point, let's talk about how do we go forward," said Tillerson.
"I think that was the right place to spend our time, rather than spending a lot of time having a disagreement that everybody knows we have a disagreement," added the secretary of state.
But Senate Democratic leader Charles Schumer ripped into Tillerson over the case.
"For Secretary Tillerson to say that this issue will remain unresolved is disgraceful -- to give equal credence to the findings of the American intelligence community and the assertion by Mr. Putin is a grave dereliction of duty and will only encourage Russia to further interfere in our elections in the future," he warned.
Climate deal 'irreversible'
Friday's encounter could have an impact on issues ranging from the North Korean crisis and conflicts in Syria and Ukraine to US-Russian disarmament treaties, world trade and global warming, analysts say.
How the "contrasting styles of machismo interact... will likely be the defining feature of their relationship," noted Derek Chollet from think-tank German Marshall Fund.
But if Trump and Putin have established a rapport, the distance between the US leader and America's long-standing allies have appeared to widen with his climate-sceptic opinions and "America First" policy.
The growing rift has turned this year's G20 summit -- normally a ripple-free event in the diplomatic calendar -- into one of the stormiest in the forum's history.
"We are not going to paper over the differences but rather, we will call discord discord," said German Chancellor Angela Merkel.
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A draft communique seen by AFP Friday appeared to reflect the 19-versus-one stance on climate.
The draft underlines that the 2015 Paris climate accord is "irreversible" and affirms that other G20 nations are committed to the deal while taking note of Washington's decision to quit the agreement.
Not given up hope
Meanwhile, North Korea's successful test of an intercontinental ballistic missile this week also casts a dark shadow over the US leader's first G20 summit.
Trump had warned Thursday that Pyongyang's military sabre-rattling would bear "consequences", saying he was considering a "severe" response to its "very, very bad behaviour".
After repeatedly urging Beijing to ratchet up the economic pressure on North Korea, Trump will hold what promises to be a testy meeting with Chinese President Xi Jinping on the G20's sidelines on Saturday.
Tillerson said the US will continue to press China to do more to rein in Pyongyang.
"Our engagement is unchanged with China and our expectations are unchanged. We have not given up hope," he added.
Russian President Vladimir Putin greets US President Donald Trump's wife Melania (AAP) Source: AAP
Unacceptable violence
Meanwhile, outside the heavily guarded G20 conference hall, protesters wreaked havoc, engaging in running battles with police in the all day as the rattle of helicopters was heard overhead.
The activists blocked US First Lady Melania Trump at her residence for hours as they torched cars, smashed shop windows, fired flares at police helicopters and even slashed tyres on vehicles belonging to the Canadian delegation.
The violence forced Hamburg police to call in reinforcements from other German states, and G20 organisers to drastically curtail an official programme for spouses of visiting dignitaries.
G20 host Merkel condemned the violence, saying she could understand peaceful protests, but demonstrations that "put peoples' lives in danger, put the protesters' own lives in danger... are unacceptable".
