One man received minor injuries but did not require hospital treatment following the incident, after which the cars were all abandoned in the street.
Any witnesses are urged to call police.
The three cars involved were a silver Fiat Stilo, a black Vauxhall Astra and a blue Vauxhall Omega.
Det Sgt Craig Hurst of Greater Manchester Police said: “This is an unbelievable incident on the streets of Atherton.
“I know the community will be alarmed but we are putting all our efforts into finding those responsible.
“There will be extra police in the area and anyone with any concerns can speak to our officers.”