Graham used a play-action fake to hit Zeron Flemister with a three-yard touchdown strike, the tight end's first of the season, to cap a 10-play, 54-yard drive and complete the comeback from a 10-point, second-quarter deficit.
"I felt we had good opprtunites. We just had to be patient and be smart with the football," Graham said.
A veteran of 10 NFL seasons and three other teams, Graham also played quarterback for the New York Giants when they ended the John Elway-led Broncos' quest for a perfect season on Dec. 13, 1998.
Graham came off the bench with 1:50 to go in the opening half after Banks suffered a concussion and left the game on a stretcher as a result of a hit from behind by Leon Lett.
"I was just trying to be as focused as possible. I was trying to take care of the ball and not worry about the next play," Graham said. "I didn't know what was going to happen. I just wanted to fit into the system and make smart decisions."
Graham completed 12 of 18 for for 123 yards as he revived a listless Redskin aerial attack. Banks hit four of 13 attempts for 27 yards.
"What they were doing was seven, eight, nine people on the line of scrimmage," Washington coach Marty Schottenheimer said. "I think their feeling was we couldn't beat them thowing the ball. We were forced to come out and do some quick short passes."
Schottenheimer, who is in his first year as Redskins coach, was dealt his most painful losses by Elway and the Broncos when he coached the Cleveland Browns and Kansas City Chiefs. He posted a combined 2-10 career record at Denver for those AFC teams.
Brian Griese and his Broncos did have a chance to drive for a tying touchdown in the final 2:37. But the Pro Bowl quarterback completed just one of seven passes on the final drive as a mixture of rain and sleet fell.
Denver fell to 5-5 and dropped 2 1/2 games behind AFC West leading Oakland.
A career 61.5-percent passer entering Sunday, Griese completed just 11 of 31 passes for 114 yards with a touchdown.
"We couldn't get anything going. Our offense was the worst performance I've seen ever in my time in the National Football League," Broncos coach Mike Shanahan said. "In the second half, our defense hung in there. We gave them great field position. We gave them a number of opportunites. We just didn't produce offensively."
Recording their first road win of the season, the Redskins (4-5) are two games out of first place in the NFC East.
Griese received limited support from Denver's running game. Terrell Davis, the 1998 NFL MVP, missed the game after undergoing a surgical procedure on his left knee earlier in the week.
Mike Anderson and Olandis Gary, who each have a 1,000-yard season to their credit, were stiffled by Washington's defense. Anderson gained 31 yards on 13 carries and Gary collected 28 yards in 10 attempts.
"We're juggling a few people in and out of there and it's hard to be cohesive," Shanahan said. "These guys are professional players. One guy makes a mistake here or there. Until we play together or play for 60 minutes we're going to struggle on offense."
Griese was forced to scramble for 29 yards on four carries. He picked up 23 early in the second quarter, moving the ball to the Washington 23. That set up Jason Elam's 33-yard field goal, the opening score of the contest, with 11:03 remaining in the half.
Stephen Davis, though, managed to keep the Denver defense honest. The Redskins running back, who entered the game as the NFC's third leading rusher, had 23 carries for 84 yards.
That helped to open up the passing game and benefitted Michael Westbrook, who had 104 yards on nine catches.
"They were blitzing, leaving their defensive backs man to man with us," Westbrook said. "It was kind of an insult. We still get no respect, even after the teams we have beaten. They just lined up and said 'let's see, mano-to-mano, who's best.'"
Graham found Westbrook three times for 35 yards on the Redskins' first touchdown drive. The seventh-year receiver caught a five-yard touchdown pass to knot the contest at 10-10.
Westbrook has tallied 15 receptions and 299 yards with two touchdowns in three career games against the Broncos.
"He moved the ball well, had a high completion percentage. It was beautiful," Westbrook said of Graham.
The slippery conditions made it difficult for any player who held the ball. Each team fumbled six times with the Redskins losing it three times and the Broncos turning it over twice.
"We kind of dug ourselves a hole, turning the ball over as many times as we did," Schottenheimer said. "I guess one thing you could say is that when you have the group of people we've got in there, with the purpose they have, then maybe they can overcome three turnovers."
"It was wet and windy out there," Graham said. "You never knew what kind of ball you were going to get -- definitely tough conditions."
Washington's Eric Metcalf failed to handle Tom Rouen's punt, which was recovered by Denver's Keith Burns at the Redskins' 13 with 6:32 left in the half. On the sixth play of the ensuing drive, Griese threw his seventh touchdown pass in the last four games, a one-yard strike to NFL receiving leader Rod Smith.
But that was his last catch of the day. Smith left after aggravating the twisted ankle he sustained in last week's win against San Diego.
With the always-dangerous Smith sidelined, Washington took control of the final 30 minutes. The teams ran a combined 64 plays in the second half, 54 of them in Denver territitory.