Witnesses said after the attack, at Forest Hill station at about 13:20 GMT, a man then chased other people outside the station.
People said they saw a man attack another on the train then run out of the station shouting "he wanted to kill Muslims".
A 38-year-old man has been arrested on suspicion of causing grievous bodily harm and remains in police custody.
Train services were temporarily stopped on the line.
Police said they were not treating the attack as terrorism, but as a hate crime.
Ambulance officials said the victim, who is in his 40s, was treated at the scene for a stab injury and taken to a south London hospital.
He remains in hospital in a serious, but not life-threatening, condition.
Emma Pinder, who was at the station, said a man had "chased other people outside the station with a knife".
Shellby Curry, 24, from Forest Hill, said she saw a man "waving what looked to be a knife in his hand" near the station and screaming words like "Muslims... kill them all".
She described the attacker as a black man with short hair, who she thought appeared to be in his late 30s or early 40s.
Several people phoned the police, who arrived about 10 minutes later and restrained him, Miss Curry said.
"I was walking along and someone stopped and said, 'be careful, he's got a knife'. As I looked around there he was.
"I just grabbed my baby and ran across the road. Looking back I saw him marching up and down waving his arm about."
Appealing for witnesses senior investigating officer Det Ch Insp, Paul Langley, said: "This was a serious assault on a man in broad daylight and we are aware that other passengers may have seen what happened or been threatened themselves.
"We have spoken with many witnesses from the train but if you saw what happened and haven't yet contacted police, please do so as soon as you can."
Services were stopped on the rail line but have since reopened with severe delays, London Overground said. Roads around the station were also shut for a while.

California's court also ordered him to pay $80,900 (Â£47,160) in damages, serve five days of community service and complete an anger management programme.
Bieber, 20, was not present in court for the sentencing.
The singer faces two other criminal cases in Florida and Toronto.
"Justin is glad to get this matter resolved and behind him," Bieber's representatives said in a statement after the verdict by Los Angeles Superior Court in Van Nuys.
"He will continue to move forward focusing on his career and his music."
Investigators searched Bieber's Calabasas home in California in January for evidence after the egging apparently caused serious damage.
One of the singer's friends was arrested on drug possession charges after that search.
The LA County Sheriff's Office say they were looking for surveillance video tapes when they spotted the drugs "in plain view".
Surveillance footage seized from the home appears to show Bieber high-fiving friends and celebrating after throwing eggs at his neighbour's home, police said.
The pop star could have faced a more serious felony charge if the damage to the home had been greater than $20,000.
He has since moved out of Calabasas to Beverly Hills.
Two previous investigations into Bieber's conduct by the sheriff's department did not lead to charges.
Prosecutors declined to charge him last year after a neighbour complained he drove recklessly through the area, and in November 2012 after a paparazzi photographer accused Bieber of assault.
He was arrested by police in Miami in January and charged with driving under the influence and driving without a valid licence and Toronto prosecutors have accused Bieber of assaulting a limousine driver in December.

The Flybe plane, which was carrying 62 passengers from Exeter, landed safely after declaring a full emergency at 13:43.
Edinburgh Airport said nobody was hurt during the incident.
A Flybe spokesman said: "The flight landed safely at Edinburgh Airport following the report of a technical fault with the aircraft."
He added: "The pilot followed standard operating procedures and, as is usual in such circumstances, the airport put its emergency vehicles on standby.
"All 62 passengers disembarked as normal when the aircraft arrived on stand."
A man has been stabbed on an Overground train in south-east London.
[NEXT_CONCEPT]
Canadian pop star Justin Bieber has pleaded no contest to misdemeanour vandalism in connection with the egging of his neighbour's home and has been sentenced to two years' probation.
[NEXT_CONCEPT]
A plane has made an emergency landing at Edinburgh Airport after having issues with its nose wheels.