Green, a cornerback, was in his first year of eligibility, while Monk was in his eighth.
Their names were called by a selection committee at the site of Super Bowl XLII in Phoenix.
They were joined by San Diego/San Francisco defensive lineman Fred Dean, Kansas City cornerback Emmitt Thomas, New England linebacker Andre Tippett and Minnesota/Denver offensive tackle Gary Zimmerman.
Green, once the league's fastest player, spent his entire 20-year career with the Redskins, holds the NFL record with at least one interception in 19 straight seasons, had 54 career interceptions with six touchdowns and was named to the Pro Bowl seven times.
"This is incredible," Green told The Washington Post. "It's just so special."
Monk once was the NFL's all-time leader in receptions, caught a pass in 164 straight games and finished his 14-year career with 820 receptions.
Among those not picked were receiver Cris Carter, who played most of his 15 seasons with Minnesota and finished second all time in both receptions and touchdown catches, and former NFL commissioner Paul Tagliabue.