Jonathan Stewart also ran for a third-quarter touchdown for the Panthers (2-5), who for the first time all year found themselves playing with the lead almost the entire way.
Newton threw for 256 yards and ran for 59 more. He ran for his seventh touchdown on a 16-yard keeper in the third quarter, tying a record for most TDs rushing by a rookie quarterback since the 1970 NFL-AFL merger.
In the fourth, he had an easy touchdown toss to Brandon LaFell that blew the game open.
Washington's John Beck threw for 279 yards and scored on a short keeper in his first start in four years. But he didn't get the Redskins (3-3) back to the end zone again until they were down 30-13.