Ambulance crews treated 21 people, including a three-year-old girl, at the scene and five were taken to the hospital although their injuries were not believed to be serious, authorities said.
The incident in the Terminal 3 garage occurred after an argument escalated between two groups who knew each other, the Metropolitan Police said.
It was not being investigated as terrorism.
One man was being held on suspicion of assault.
Police were searching for other suspects who left the scene.
A 31-year-old man who was arrested on suspicion of assault remained in custody on Sunday afternoon, police said.
Officers said the man was arrested by 8.20am, within nine minutes of the report being received.
The incident caused several hours of delays for passengers using the terminal.
In a statement on X, Heathrow Airport said its teams were also responding to the incident, with passengers advised to allow extra time when travelling to the airport and to check with their airline for any queries.
Witness Tom Bate told the BBC he had been waiting to be picked up from the car park when he saw "young men dressed in black darting through the crowd".
He said he felt a burning at the back of his throat after they left, with other people nearby beginning to cough.
with PA
26°C