It was the Seahawks' first play-off victory on the road since a 1983 win over the Miami Dolphins, and set up a divisional round clash against NFC South champions Atlanta on January 13.
The victory was Seattle's sixth in a row and brought NFC East champion Washington's seven-game winning streak to an end.
Redskins' quarterback Griffin injured his knee trying to gather a bad snap and took himself out of the game in the fourth quarter.
"What was really exciting for us is that we weathered the storm, and the crowd and the situation and the setting, and our guys stayed focus and started to put the game in order," said Seattle coach Pete Carroll. "Incredible effort by the guys."
Griffin, the number two overall pick in the 2012 NFL Draft, led the Redskins to two touchdown drives on their first two possessions of the game.
Seattle's defense stiffened after that to shut down the Redskins and a limping Griffin, who wore a brace on his right knee and showed the effects of the injury as the game went on.
Wilson, meanwhile, combined with running back Marshawn Lynch to dominate the game. He rallied the Seahawks to 14-13 by half-time and added 11 fourth-quarter points to seal the win.
Lynch gained 132 yards on 20 carries and put the Seahawks ahead for good with a 27-yard touchdown run midway through the fourth quarter.
Wilson completed 15-of-26 passes for 187 yards, while Griffin was 10-of-19 for just 84 yards. Wilson added 67 yards on eight runs, and Griffin gained just 21 yards on five carries.
"I don't think you worry about the first quarter. You focus on the next play you have, stay in the now," Wilson, selected with the 75th pick of the draft, told reporters.
"The key is to get a huge win at the end of the game and we did that tonight."
Washington clung to their one-point lead, helped by a Lynch fumble at the Redskins' one-yard line in the third quarter, but gained just 63 yards after the intermission to 199 for Seattle.
I'm very disappointed today," said Redskins coach Mike Shanahan. "You always want to play your best football during the playoffs. We probably had our best first quarter, but after that first quarter we just didn't seem able to get things done."
Griffin, lacking the dynamic running burst that gave him the NFL record for rushing by a rookie quarterback this season, left the game after he could not get up from the turf to recover a botched snap deep in his own territory.
"I'm the quarterback of this team. My job is be out there if I can play," explained Griffin, who had trouble evading the pass rush and was sacked five times.
"The only time I couldn't play was when I went down, and I took myself out of the game. That's just the way you have to play it," said Griffin, who will have tests on his knee on Monday.
Washington trampled through a Seattle defense that led the NFL in fewest points allowed this season with ease at the start, as Griffin engineered drives of 80 and 54 yards ending in four-yard touchdown passes to Evan Royster and Logan Paulsen.
The Seahawks fought back in the second quarter with a pair of field goals and a 73-yard drive capped by a four-yard touchdown pass from Wilson to fullback Michael Robinson.
Lynch's late touchdown put Seattle ahead 19-14 and tight end Zach Miller caught a Wilson pass in the end zone for a two-point conversion that made it 21-14.
After recovering the fumbled snap on the Washington five-yard line, Seattle settled for a 22-yard field goal that made it 24-14. Redskins back-up Kirk Cousins replaced Griffin for the last two series but could not engineer a comeback.