The incident happened at an address in Deodar Road just after 3.30am today (January 24).
Officers and London Ambulance Service crews attended and found a man who had been stabbed.
He was pronounced dead at the scene shortly after.
Although formal identification is yet to take place, his next of kin have been made aware.
A special post-mortem examination is due to take place today.
A 30-year-old man was arrested on suspicion of murder, he remains in police custody.
It is believed he was known to the victim.
Detective Superintendent Amanda Mawhinney from the South West Command Unit which covers Putney said: “We are currently supporting the family of a man who was sadly killed in the early hours of this morning.
“I am aware that people in the community may feel shocked following this – we believe this to be an isolated incident and there is no wider threat to the public.
“Although we have made significant progress by making an arrest, I would like to make it clear that our investigation does not stop here.
“We need the local community to help us understand what happened in the early hours of this morning.
“We are appealing for anyone who was in the area at around 3am and that saw or heard anything unusual to contact us as soon as possible.
“You may notice a higher police presence within the area today whilst we carry out our enquiries.
“A scene is in place as well as road closures. If you feel the need to raise anything with our officers, then please feel free to speak with them whilst they are in the area.”
Police enquiries remain ongoing.
Anyone with information which could assist with the investigation is asked to call 101 stating CAD721/24JAN.
Alternatively you can contact the independent charity Crimestoppers anonymously on 0800 555 111 or by submitting an online form.
