A source close to Home Secretary Priti Patel branded Ben and Jerry's "overpriced junk food" after the ice cream company criticised Britain's stance on migrants crossing the Channel to the UK.
Addressing Ms Patel, the official Ben and Jerry's Twitter account said: "We think the real crisis is our lack of humanity for people fleeing war, climate change and torture."
It added: "People wouldn’t make dangerous journeys if they had any other choice. The UK hasn’t resettled any refugees since March, but wars and violence continue. What we need is more safe and legal routes."
Ms Patel has described the crossings as "shameful" and said migrants attempting to enter the UK via the English Channel should be returned to France.
Ben and Jerry's said: "Let’s remember we’re all human and have the same rights to life regardless of the country we happen to have been born in.
"People cannot be illegal."
More than 4,000 migrants have now reached the UK this year by crossing the English Channel in small boats.
In response to the tweets, a Home Office source told the BBC: "Priti is working day and night to bring an end to these small boat crossings, which are facilitated by international criminal gangs and are rightly of serious concern to the British people.
"If that means upsetting the social media team for a brand of overpriced junk food, then so be it."
On Monday, Prime Minister Boris Johnson branded migrants’ Channel crossings a "very bad and stupid and dangerous and criminal thing to do".
Ms Patel, who went to see for herself the Border Force operations at Dover, said the UK authorities now needed to work with the French to make the route “unviable”.
“The number of illegal small boat crossings we have seen recently is totally unacceptable,” she said.