A 34-yard field goal by Kai Forbath, set up by rookie Richard Crawford's 64-yard punt return, ended the game four minutes into the overtime and ignited the crowd of 81,178.
The victory, however, belonged to Cousins, subbing for Robert Griffin III after the rookie starter suffered a knee sprain with 1:56 left in regulation on a tackle by the Ravens' Haloti Ngata.
Cousins tossed an 11-yard scoring strike to Pierre Garcon with 29 seconds left in regulation and then successfully ran up the middle for the two-point conversion to force the overtime.
The rookie from Michigan State said the final minutes of the fourth quarter "all happened so fast I am not sure I can replay it all correctly until I watch the tape.
"I thought, 'I am going to make something happen and try to do my best RG3,'" said Cousins, who completed his two passing attempts for 26 yards.
"Pierre ended up coming off his route and breaking away, breaking open - wide open - and all I had to do was put it his arms."
The Ravens (9-4) squandered an opportunity to secure a playoff berth but still hold a two-game lead over Pittsburgh (7-6) and Cincinnati (7-6) in the AFC North.
Baltimore, which lost to Pittsburgh 23-20 last week on a last-second field goal, was hampered by the loss of linebacker Terrell Suggs, who sat out the game with a torn biceps.
"We all need to do better," said frustrated Ravens head coach John Harbaugh. "We all need to find a way to finish the game better. That's the bottom line."
Ravens quarterback Joe Flacco completed 16 of 21 passes for 182 yards and threw three first-half touchdowns, two to Anquan Boldin and one to Dennis Pitta.
Baltimore's Ray Rice had 121 yards on 20 carries and a touchdown, almost the same as Redskins rookie Alfred Morris, who totaled 122 rushing yards on 23 attempts and a score.
The triumph, the Washington's fourth straight, lifted the Redskins' record to 7-6 and on the heels of the front-running New York Giants (7-5) in the NFC East. New York faced New Orleans later Sunday.
"It takes everyone to win a game like that - all 46 guys that were dressed," said Redskins head coach Mike Shanahan. "We played extremely hard and it was a nice win for us."
Griffin, the Redskins' rookie sensation, completed 15 of 26 passes for 246 yards and a touchdown but was harassed much of the day by the Ravens defensive front.
After getting hurt and leaving the game, Griffin returned for four plays but was hobbling badly and had to go back to the sideline.
The extent of the injury was not immediately known and Griffin, affectionately known as RG3, said he was not worried. A magnetic resonance imaging (MRI) exam on Monday should determine the seriousness of the injury.
"I am not a doctor but I know what an ACL (torn anterior cruciate ligament) feels like and it doesn't feel like an ACL," said Griffin. "So hopefully we can have some positive things come out of the MRI.
"We took an X-ray and it was a very good X-ray. I feel the positive vibes coming my way and hopefully everything will be all right."
Just 40 miles (64km) separate the cities of Baltimore and Washington but the two teams have not played since 2008 when the Ravens beat the visiting Redskins 24-10.
Baltimore hosts the Denver Broncos (10-3) next Sunday, while the Redskins visit the Cleveland Browns (5-8).
Morris said he did not feel the Redskins will face a letdown against the Browns.
"It's the NFL," he said. "Any level of football, on any given, day, you can be beaten. I don't care if it's the worst team or the best-record team. You have to bring your 'A' game.
"You have to play any team like it's your last."