Police were called to a house in Byron Crescent at about 16:30 GMT where a man in his 30s was found. He died in an ambulance on his way to hospital.
A post-mortem examination will be held in due course but police confirmed he had suffered stab wounds.
The arrested man was being questioned on Monday night at Kempston Police Station.
Detectives are appealing for anyone with information about the incident to contact them.
A man in his 40s has been arrested on suspicion of murder after the death of a man in Bedford.