The Tory MP said there would be a "mucky compromise" with Russia had the UK remained tied to Brussels – as the bloc's principle of "sincere cooperation" would force it to "follow a Franco-German line in dealing with Russia, which is what we did in 2014".
Appearing on Sky News's Sophy Ridge on Sunday show, the backbencher also attacked the government for scaling back its plans to scrap thousands of EU laws retained after Brexit.
Rees-Mogg claimed the UK had saved £191bn by not being part of the EU's COVID recovery fund, and that its ability to strike free-trade deals, such as the one with Australia, would boost Britain's economy.
The former Brexit opportunities minister said: "We need to understand the point of getting rid of retained EU law – it's not about whimsical euro-scepticism, it's about making the economy better while we're in an inflationary period."
Watch: Brexit stopped Ukraine invasion from succeeding, Jacob Rees-Mogg says
He added: "One of the ways to bring down prices is by tackling monetary policy. The Bank of England is doing that and that's difficult for people.
"The other thing you can do that brings prices down is free up the economy from over-regulation."
Rees-Mogg has attracted criticism over his claims made on Sunday morning, with political economist and columnist Will Hutton tweeting: "The Brexit fantasy has no limits.
"Here is Jacob Rees-Mogg claiming that Brexit stopped Russia from succeeding in Ukraine because it allowed UK freedom to lead. US leadership, money and arms? Nato? Would EU prevent UK action? He lives on planet Bonkers."
Liz Webster, chair of the pro-single market Save British Farming campaign, wrote: "Desperate to show a #Brexit upshot, joker Rees Mogg makes ludicrous and insulting claim about #Ukraine invasion."
War journalist Oz Katerji suggested quite the opposite to what Rees-Mogg said, tweeting: "Look no matter which way you voted on Brexit, it's objectively true Russia threw its weight behind Brexit as a means of undermining the European Union, and no amount of Rees-Mogg b******t changes that."
He added: "Brexit doesn’t have to be a 'Russian plot' to make that true, but we can all surely agree that, as absurd as that argument is, it's nowhere near as absurd as Rees-Mogg claiming Brexit thwarted Putin's invasion of Ukraine. This is a fundamentally ridiculous man unfit for office."
Rees-Mogg's latest remarks on the economy appear to be at odds with his claim last summer that the cost of living crisis has "very little to do with Brexit".
At the time, he told LBC radio's Tonight With Andrew Marr show: "These are headwinds faced across the world, inflation is running across the world.
"The government is tackling a global, economic problem... Supply chains are not working across global economies and this is to do with lack of Chinese semi-conductors and cities in China still being closed. It is to do with shipping being in the wrong place, this is very little to do with Brexit."
Also appearing on Ridge's show on Sunday was former Labour spin doctor Alastair Campbell, who had plenty to say about Rees-Mogg's claims.
"I would definitely say the UK has done a good job in relation to Ukraine, but that has nothing to do with us being outside the European Union," he said.
"The Australian [post-Brexit] trade deal, and all the other trade deals – they've cut and pasted what we had with the EU – all they're adding is a miniscule amount in the long-term to our trade, which has meanwhile been absolutely battered."
Campbell referred to The Sovereign Individual, a book written by Rees-Mogg's father William in 1997, claiming "that is what Brexit it about for the Jacob Rees-Mogg types... by getting rid of taxation, getting rid of regulation and actually getting rid of democratic control.
"If that's the best he can do on the benefits, given he had the job of secretary of state for Brexit opportunities, it underlines to me the fact that Brexit is going very, very badly wrong."
Defending the government's climbdown on the Retained EU Law Bill, energy secretary Grant Shapps told Ridge: "The reality is, as the libraries look for more of these laws that were sort of almost leftover pieces of law, more are being discovered in the archives, as it were, and so there's a practicality for simply getting those through.
"We're now several months on from when I was business secretary and looking after this process. So more of the laws will have been uncovered because that was the work that was going on.
"I have every confidence the current business secretary is looking at the plethora of these laws that are left, she's going to have more than 2,000 scrapped by the end of the year. She just announced another 600, I think she's doing this the right way."