The man is seriously injured in hospital after the incident in Nowell Lane, which happened at 4pm on Tuesday.
The woman, who did not want to be named but lives nearby, said: "I heard shouting and screaming.
"I ran outside but at first I didn't even realise it was him."
The woman said she was friends with the man and that he is 41.
She said she shouted to her son to get blankets as they tried to tend to his wounds.
The woman added: "I don't know what happened before this. The man is my friend.
"He was bleeding from anywhere and everywhere. No one deserves that."
Forensics officers were taking photographs of bloodied rags on the pavement nearby later in the evening.
Photographs were also being taken of a Volvo in Kimberley View.
There was a heavy police presence in the area throughout the evening as detectives continued their investigations.
A stretch of Harehills Lane was cordoned off around the junction with Nowell Lane, with traffic diverted down Nowell Walk.
Several police cars were also stationed in Nowell Street.
A 45-year-old man was arrested on suspicion of affray, but enquiries are ongoing, a West Yorkshire Police spokesman said earlier.
Anyone with information should contact police on 101 or call Crimestoppers anonymously on 0800 555 111.