The Redskins never trailed and stopped Arizona on downs at the Washington 21-yard line with 16 seconds left.
Washington was slightly better on a day when neither offense was impressive.
Arizona went 0-for-6 in the red zone and was 4 of 19 on third down. The Cardinals haven't scored a touchdown in their past two games and were officially eliminated from the playoff hunt.
Cousins was 18 of 26 for 196 yards and no interceptions. Washington (6-8) snapped a two-game losing streak and won despite rushing for just 32 yards.
Blaine Gabbert finished 17 of 41 for 197 yards for Arizona. He threw an interception and fumbled twice, losing one that led to a Redskins' touchdown.
Phil Dawson kicked five field goals without a miss one week after hitting four in a 12-7 win over Tennessee.
The Cardinals (6-8) haven't won back-to-back games this season
Washington linebacker Preston Smith had an interception, fumble recovery and two sacks.
Arizona pooched the second-half kickoff and, when no one from the Redskins got there, Budda Baker of the Cardinals recovered at the Washington 22. The Cardinals could not move the ball however, and Dawson's fourth field goal pulled them within 14-12.
Later in the third quarter, Washington started on its 6-yard line and Cousins passed for 67 yards in an 88-yard drive before settling for a 24-yard field goal by Dustin Hopkins.
Dawson answered with a 54-yarder early in the fourth quarter, but a leveraging penalty on the Redskins gave the Cardinals new life on the Washington 20. Another red-zone failure resulted in Dawson kicking a 32-yarder to pull Arizona within 17-15 with 11:13 left to play.
Hopkins restored the five-point lead from 32 yards out with 4:30 remaining.
Arizona got the opening kickoff and turned the ball over three plays later when Gabbert was sacked by Anthony Lanier and fumbled. Smith recovered at the Cardinals' 18 and made it to the 6. Cousins found Jamison Crowder on a crossing pattern for the touchdown.
The Cardinals answered with a 16-play, 60-yard drive that stalled on a pair of sacks and ended with Dawson's 40-yard field goal.
Washington made it 14-3 early in the second quarter when Cousins threw a screen pass to running back Kapri Bibbs, who scampered 36 yards for a touchdown. Bibbs was signed off the practice squad this week after Byron Marshall was placed on injured reserve.
Trailing 14-6, the Cardinals drove to the Redskins 9 late in the half, but on first-and-goal, Gabbert's pass was off the mark and Smith picked it off.
On first down at the Redskins 41, Cousins hit tight end Vernon Davis, but he fumbled and Arizona's Tramon Williams returned it to the 33. The Cardinals reached the 2, but after three incomplete passes, settled for another Dawson field goal on the final play of the half.
NOTES: Cardinals LB Chandler Jones became just the third player in franchise history with 15-plus sacks in a season. ... Arizona WR Larry Fitzgerald (five catches) reached the 90-catch plateau for the eighth time in his career. No other player in NFL history has more than six such seasons. ... Washington CB Bashaud Breeland left with a shoulder injury in the fourth quarter. ... Redskins K Dustin Hopkins was activated off injured reserve on Saturday and the team waived K Nick Rose, who went 10 of 11 on field goals and 18 of 20 on extra points in Hopkins' absence. ... Washington LB Zach Brown missed the game with multiple injuries. T Trent Williams (knee) and DL Terrell McClain (toe), listed as questionable during the week, were out. ... Arizona TE Jermaine Gresham (illness) and DT Josh Mauro (ankle) were out after being listed as questionable.