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.
A man has been stabbed on an Overground train in south-east London.