Jones, a third-round draft pick out of Florida, scored from 39 yards in the first quarter, and from 3 with about 2 1/2 minutes left in the fourth.
The Redskins (1-1) led 17-0 at halftime, the first time they shut out an opponent in the first half since Oct. 2, 2011, also against the Rams (1-1).
Cousins went 23 for 27 for 203 yards and a 4-yard TD pass to Pierre Garcon. Most importantly, he did not throw an interception.
Robert Griffin III was inactive for Washington, reduced to No. 3 quarterback status.