Saying he didn't know where to begin with all the allegations, Zakaria began by asking Peskov directly if the Russian government had "any collaboration or serious communication back and forth with Donald Trump's campaign during the election campaign last year."
"The answer is very simple, no," Peskov replied.
"The fact that Russia is being demonized in that sense comes very strange to us. And we are really sorry about that," he added.
What's more, he said, America was "self-humiliating" by admitting that a foreign country had the power to intervene in its election process.
"America," Peskov said, is "a huge country, a country number one, the most powerful country in the world, with a very, very stable political tradition and you say that a country can easily intervene and easily influence your electoral process. This is simply impossible."
Russia's stance on this is simple, he said.
"We will never let anyone to put his nose in our domestic affairs, but we're quite reciprocal in that attitude when it comes to other countries."
Asked by Zakaria what he made of the fact that "all 16 United States intelligence agencies say they have high confidence that Russia tried to alter the outcome of the elections in the United States," Peskov said he had not seen anything "trustful."
"We have never seen any evidence," he said.
"What we have seen in open, a public part of a report by one of the agencies, special agencies, of the United States, and I would humbly say that it's not a paper of a high quality in terms of being really trustful," he said, referring to the heavily censored January intelligence report into Russian cyberhacking activities during the 2016 campaign.
"He was talking about bilateral relations," Peskov said.
"He was talking about what is going on in the United States so we have a better understanding in Moscow. This is what is being performed by every ambassador of Russia abroad, every ambassador of the United States abroad, including in Moscow."
Challenged by Zakaria as to whether Kislyak had any similar meetings with Hillary Clinton, Peskov was vague.
"If you look at some people connected with Hillary Clinton during her campaign, you would probably see that he had lots of meetings of that kind," he said.
However, he was keen to emphasize, "There were no meetings about elections -- electoral process ... in no way should it be presented as interference in electoral process."
Anyone who might assume otherwise, he said, was simply looking at the situation "with intent to demonize Russia."
Asked his thoughts on the first weeks of the Trump administration, Peskov said his boss, President Putin, had found President Trump pragmatic, but also expressed some concern.
"We don't have a proper understanding of perspectives of future of our bilateral relationship," he said.
"We certainly would expect our contacts to be more frequent, more in-depth, in order to sit and then talk to each other to try to understand, because we had quite a significant pause in our bilateral relations," he said.
"We are losing potential by blaming everything on Earth on each other."