Heavy rainfall in southwestern Connecticut led to mudslides, washed-out roads and flash flooding on Sunday, with some towns reporting as much as nine inches of rain.

The National Weather Service declared a flash flood emergency for parts of Fairfield and New Haven counties, as overfilled rivers crested their banks and additional thunderstorms were predicted. Emergency crews carried out widespread water rescues, especially in the Southbury area, and several mudslides were reported, according to the National Weather Service.

Rivers breached roadways in several places in and around Southbury and Newtown, with some images on social media showing cars submerged up to their windshields. Chris Grillo, the police chief of Southbury, said that there had been no injuries reported, though the police had helped several people out of flooded vehicles.

The Connecticut Department of Emergency Management warned of a severe risk of flooding around Southbury into the evening. Several roads were also closed in the Danbury area.

“Please avoid flooded roadways and turn around don’t drown,” the department urged on social media.

Elizabeth Savio, 23, a team manager at Panera Bread in Southbury, said Sunday afternoon that the intersection in front of the shopping center, which includes the store, had completely flooded, and that four cars had become submerged in the floodwaters.

She said that she had watched as a man swam to one of the cars to rescue the driver, and that she had helped the two of them reach Panera. The rescuer, Lucas Barber, then swam back to the car to retrieve the man’s prosthetic leg and his golden retriever.

Mr. Barber, 30, said: “I pulled over, grabbed some rope that’s in the back of my car for emergencies, and threw my phone and wallet on the seat and ran out there.” He said he had checked the other nearby cars to ensure they were empty but had seen that Patrick Jennings, the man he rescued, was still inside his vehicle.

Mr. Jennings called him a “savior.”

“He got me all the way out of the water, the dog comes swimming up and the rest is history,” Mr. Jennings said.

In Newtown, the police department was overwhelmed with calls about non-life-threatening flooding and asked residents to call only in a true emergency, the department posted on Facebook. There were no immediate reports of injuries there.

“We understand that water in your basement is alarming and inconvenient, however, emergency resources are not equipped or staffed to respond to minor basement flooding,” the police department wrote.

Some Metro-North trains on the Danbury and New Haven lines were delayed, skipped stops or were replaced with buses because of flooding through the area.

While training thunderstorms are common in the summer, he said, rainfall totals as high as these are not: “This amount of precipitation wasn’t expected by anyone today, though we were expecting rain.”

“Generally predicting stuff like this is difficult,” he said, adding that the phenomenon can be very localized, as wind patterns both at the surface and aloft keep the thunderstorm in the same place.