They were targeted by two men on a motorbike on Saturday morning in an attack caught on CCTV.
Officials said the holidaymakers were shot in the struggle as they tried to resist attempts to steal their belongings.
Both were taken to hospital after the attack near the entrance of the Faena Art Hotel, in the waterfront Puerto Madero district of the Argentinian capital.
A 50-year-old man who was shot in the chest later died, while his 28-year-old stepson suffered a thigh wound.
Local media reports said police believe the motorbike robbers were supported by accomplices in a vehicle and are investigating whether the victims were followed to the hotel from the airport.
A Foreign Office spokeswoman said: "We are supporting the family of two British men following an incident in Buenos Aires, and are in contact with the local authorities there."
More than 111,000 British nationals visited Argentina in 2018, according to the Foreign Office, which said most visits are "trouble free".
But tourists are warned to be alert to street crime, including armed robberies, and advised to hand over cash and valuables without resistance.