No injuries were reported.
"We could feel the plane sliding off the runway and then (it) came to a very sharp halt" after landing, CNN producer Elizabeth Landers, who was on the plane, told the channel.
"The governor and everyone on board is okay," she said of Donald Trump's running mate.
The Boeing 737 slid completely off the runway onto grass next to the tarmac, she added, saying there was mud and grass on the plane's body and "gashes" on the runway.
Video footage broadcast from the runway soon after the incident showed steady rain falling. Pence could be seen shaking hands and posing for photos with first responders.
"So thankful everyone on our plane is safe," he tweeted after the incident. "Grateful for our first responders & the concern & prayers of so many. Back on the trail tomorrow!"
Trump, who was campaigning in Ohio, called Pence after the accident, spokeswoman Stephanie Grisham said.
"He reached out to Governor Pence and he is very glad everyone aboard is safe," she said.
Trump's Democratic rival Hillary Clinton also tweeted after the incident: "Glad to hear @mike_pence, his staff, Secret Service, and the crew are all safe."