Police were making door-to-door enquiries in Wilberforce Road this morning.
They were both taken to an east London hospital, but the boy was pronounced dead shortly before 1am. The girl remains in hospital in a critical condition, the police said.
Det Ch Insp Dave Whellams, of the homicide and major crime command, said: "This is clearly a tragic incident and our thoughts are with the family of the two young children". He is Bidhya Sagar Das.The incident is believed to be domestic.
The Metropolitan Police said the child's next of kin are aware and that formal identification and a post-mortem will be carried out. After appealing for help in tracking down the suspect, they arrested him not far from the scene of the incident on Sunday evening.More news: Novak Djokovic joins Andy Murray, Serena Williams in Miami withdrawal
The National Police Air Service helicopter tweeted that it was assisting "officers searching Finsbury Park for a suspect".
Neighbors told the BBC that a woman, believed to be the children's mother, rushed out of the flat late Saturday night screaming for help and calling, "My kids, my kids!". I heard someone was shouting.
Mihai Manea, 29, who lives on the second floor of the building, said the children were twins and lived on the top floor.
"She kept on shouting". "There's police coming and going all the time". "(We are) just shocked, you wouldn't expect something like this...when I saw the police, I said, 'It's just Saturday night, it's like, somebody got drunk or in a fight but I wouldn't imagine like a murder".