The four victims were three women aged in their mid-60s, 40s and 30s, and a man in his mid-60s, police said.
The niece of one of the four Bermondsey stabbing victims says her aunt "does not deserve this.”
A man in his 20s has been arrested on suspicion of murder after three women and a man were stabbed to death at a south London home early on Monday morning.
The victims were all pronounced dead at the scene in Delaford Road, Southwark.
The Met Police say it is believed all five people were known to each other, and detectives are not looking for anyone else in connection with the incident.
Police at the scene said the victims were believed to be aged in their mid-60s, 40s and 30s, and one man believed to be aged in his mid-60s.
Venecia Reid paid tribute to her "kind" and "generous" aunt near the scene of the tragedy.
The victim's niece told reporters: “She was very loving, very kind, very generous.
“She worked very hard in this country to support her two girls and her two grandkids.
“She does not deserve this.”
The victims have not yet been officially named, and police have urged the public to avoid speculating about the incident as homicide detectives launched a probe.
The Met said officers were called by neighbours at about 1.40am Monday, 25 April, to reports of a disturbance at the residential address.
Officers forced entry and found four people suffering what were believed to be stab injuries.
Despite emergency services' efforts, all four were pronounced dead at the scene.
Chief Superintendent Colin Wingrove said: "This is a terrible incident and I fully understand how worried the community will be. We are working closely with our partners and local community groups to keep everyone informed and answer their questions."
"Extra officers will be in the area to offer reassurance - please speak to them, they are there to listen to you and act on your concerns."
"Homicide detectives have launched a wide-ranging and relentless investigation into these terrible events, please assist them by avoiding unhelpful speculation and allow them time to uncover the evidence."Such terrible events are rare, but I know that will not diminish the shock that will be felt across London and the wider country. I assure you, this investigation will leave no stone unturned in discovering the facts."
Part of the residential street was closed off with metal hoarding and a number of forensic tents had been put up as police carried out their investigations on Monday.
Forensics officers were seen entering the fenced-off section at the property, which is around five minutes' walk from Millwall FC's The Den stadium.
Anne Birkett, 60, whose back bedroom looks on to the house where the victims were found, said she was woken by police sirens and a circling helicopter just before 2am.
Ms Birkett said: “It’s devastating – you hear all of this but you never assume it’s going to be right on your own doorstep.
“I’m a foster carer and I’ve got children in my care, and now I’ve got to protect them now because of what’s happened.”
The London Mayor said: “I am heartbroken that three women and a man were killed last night in a devastating incident at an address in Southwark.
“My thoughts are with the family and friends of those who have lost loved ones in this awful crime.
Mr Khan added: “I am in contact with the leadership of the Metropolitan Police.
“An investigation is under way and one man has been arrested. I urge anyone who has information that could be relevant to contact the police immediately.”
MP for Camberwell and Peckham Harriet Harman called on the community to help police
Local MP Harriet Harman described the fatal stabbing of four people in Bermondsey as a "terrible tragedy".
The MP told reporters at the scene: "Tragically four people have been killed and that is shocking to happen anywhere but especially in this very peaceful, quiet, very settled community here.
"The police have arrested someone, but it's really important still that people give information, any information they've got to the police so that whoever committed this horrific crime can be brought to justice".
Asked about the victims, Ms Harman said: "I don't really want to say anything more than the police are actually saying.
"The police will be giving information and are obviously conducting the investigation with all the forensics here as we speak.
"But it's a terrible tragedy and our thoughts are with the family and friends of those who've tragically had their lives cut short".
The Met is urging anyone with information about the incident to contact officers on 101 or contact via Twitter @MetCC, quoting CAD 406/25APR.
Alternatively, you can ring Crimestoppers anonymously on 0800 555 111.