The Sunday night incident was partially captured by the dashboard camera of a police cruiser that responded to the scene in Willoughby, where the suspects' vehicle reached speeds of up to 60 mph as the restaurant worker clung to the hood.
The fleeing car eventually got stuck in the snow.
The 16-year-old driver and her two friends, both 17, were arrested, police said.
"I saw our manager kind of taking off through the front door, saying, 'I'm going to call police,'" said Kayla Sherman, a restaurant employee. "It caught me off guard, so I went running out there with him."
Willoughby Police Detective Lt. John Begovic said, "Noting surprises me anymore.
"Just when you think you've seen it all, something else comes up," the lieutenant said. "Again, we're just grateful that there was nobody seriously injured in this case."