The man and woman picked up their groceries with their faces wrapped in red bandannas featuring the Nazi symbol.
In footage of the incident, shared on Facebook, the woman raises both middle fingers when confronted by an onlooker.
“You're sick! Yeah, you're sick,” a man’s voice can be heard saying.
“You can't be American and wear that mask. We literally had a war about this."
The encounter took place in the town of Marshall, Minnesota on Saturday – the first day of a statewide enforcement of mandatory face coverings in shops.
The video was posted by vicar Raphaela Mueller, 24, who had visited the store with her partner Benjamin Ruesch.
Describing her shock at the couple’s deeply offensive attire, she said: “I audibly gasped.”
Ms Mueller told CBS News that after spotting the pair she proceeded to the customer service desk to inform management. By that point, the pair were checking out.
When Mr Ruesch challenged them over their “sick" behaviour, the woman suggested the act was a political statement ahead of the US presidential election.
"If you vote for (Joe) Biden you're gonna be living in Nazi Germany, that's what it's going to be like,” she said.
“Socialism is gonna happen here in America.”
When Mr Ruesch continued with his condemnation of their actions, the woman replied: "You're not getting it, I'm not a Nazi."
"Then why are you wearing the mask?" Ms Mueller asked.
"Because I'm trying to tell you if you vote for Biden this is what you are going to have – socialism."
The local police were called and a spokesperson told CBS News that the 59-year-old man and 64-year-old woman had been issued "trespass notices”.
This means they will not be allowed in any branch of Walmart for at least a year.
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The supermarket chain said in a statement: "What happened today at our store in Marshall, MN is unacceptable.
“We strive to provide a safe and comfortable shopping environment for all our customers and will not tolerate any form of discrimination or harassment in any aspect of our business.
“We are asking everyone to wear face coverings when they enter our stores for their safety and the safety of others and it's unfortunate that some individuals have taken this pandemic as an opportunity to create a distressing situation for customers and associates in our store."
Walmart said a local manager tried to offer the couple disposable face masks but they declined.