Police were called to Hartington Street, in Barrow, at 18:30 BST on Tuesday, following reports of a man causing a disturbance.
A Taser was used during the arrest of the man, who was in his 20s, and he later complained of feeling unwell. He was taken to hospital, where he died.
The incident has been referred to the Independent Police Complaints Commission.
A Cumbria Police spokesman said: "Neighbourhood police officers attended the scene and arrested a male on suspicion of causing criminal damage and, during the arrest, a Taser was deployed.
"The man became unwell following the arrest and was taken to Furness General Hospital by officers.
"At around 9pm the man, who was in his 20s and lived locally, was pronounced dead."