Redskins general manager Scot McCloughan announced the team's decision Monday, saying there is a "very positive feeling in the building right now."
Griffin will make $16.2 million in the option year in 2016. He will make $3.27 million in base salary this season.
The Redskins had until May 3 to pick up the option, but McCloughan said that the team had made its decision on Griffin "a while back."
"He's a good football player," McCloughan said. "He's got the tape out there. Everyone knows what he did in  when he was offensive rookie of the year. This offseason will be his second in the system, being healthy, being able to go through the offseason."
The $16.2 million is guaranteed for injury only, meaning Washington could release Griffin prior to the start of the 2016 league year without any salary cap ramifications.
But the Redskins have publicly supported Griffin throughout the offseason despite the former Heisman Trophy winner's disappointing first year under coach Jay Gruden. Griffin, Kirk Cousins and Colt McCoy all started at times last season, when Washington went 4-12.
Despite previously indicating that he would hold an open competition for the starting quarterback job, Gruden reversed course at the NFL scouting combine in February, saying the Redskins would enter the season with Griffin "as our No. 1 guy."
McCloughan acknowledged Monday, however, that the Redskins' decision on Griffin's option will not necessarily prevent them from selecting a quarterback in this weekend's NFL draft. Washington has the fifth overall pick.
"It does not affect our draft process whatsoever," he said. "If we get to the fifth pick and the best player's a quarterback, we're taking a quarterback. We like the three we have, but again, we're not going to pass on a good football player. There's no reason to."
Griffin appeared in just nine games last season due to an ankle injury, passing for 1,694 yards and four touchdowns with six interceptions. He has battled injuries and widespread scrutiny since his outstanding rookie season in 2012, when he led the Redskins to the NFC East title.
ESPN.com Redskins reporter John Keim and The Associated Press contributed to this report.