On 3rd-and-21 from the Dallas 30, Romo scrambled to his right to avoid the blitz and connected with Bryant in the middle of the field. A facemask penalty on cornerback DeAngelo Hall made it a 45-yard play.
"They sent everybody. I tried to give ground," Romo said. "It was a good job by Dez continuing to play on. Obviously, it was a major play in the game."
After three runs from Tashard Choice, Bailey connected from 40 yards away.
Romo connected on 22-of-36 passes for 255 yards and was picked off once. Felix Jones gained 115 yards on 14 carries.
"It was a tough game, obviously for the pain side of it for me and a lot of guys," Romo said. "It was a grind-it-out kind of game. I thought the guys did a great job battling."
Grossman went 22-of-37 for 250 yards with a TD and an interception.
"We had our opportunities there in the fourth quarter, and obviously didn't get it done," Redskins coach Mike Shanahan said.
The Redskins were trying to start a season 3-0 for the first time since 2005, but couldn't hold on against a team that was battered and was coming off an overtime win at San Francisco.
Miles Austin didn't play for the Cowboys because of a hamstring injury, but Romo had a gutsy performance in helping Dallas win a home opener for the first time since moving into its new stadium.
Graham Gano kicked a 46-yard field goal to cap the opening possession, a 10- play, 42-yard drive, but the Cowboys got the score right back on Bailey's 41- yard field goal in a march kept alive by a pass interference penalty on Kevin Barnes.
There was a key challenge by Shanahan later in the opening quarter, and it worked in Washington's benefit, leading to Gano's 27-yard field goal with 21 seconds left. Dallas faced a 2nd-and-14 from its 22 when Romo hit Kevin Ogletree with a short pass. LaRon Landry stripped the ball from behind, but the Cowboys were initially given the ball back as officials ruled Barnes touched the ball as he was out of bounds. The call was reversed when replays showed Barnes was actually in the air when he touched the ball, giving Josh Wilson the recovery at the 10.
Bailey's 27-yard field goal early in the second came after Grossman was picked off by Lee, and Dallas' Gerald Sensabaugh blocked a field goal try by Gano midway through the second.
The led to a Bailey 32-yard field goal and a 9-6 Dallas lead, but Gano split the uprights from 50 yards away to end the half.
Tim Hightower caught a one-yard TD pass from Grossman with four minutes left in the third. That score came following a turnover as Romo was picked off by Barnes.
Bailey was true from 41 yards away in the closing minute of the third and his 23-yard boot with 6:58 remaining made it a one-point game. Jones ran for 40 yards and Romo found Laurent Robinson for a 25-yard pass play on the drive that got Dallas within 16-15.
The Cowboys own a 35-16-1 record in home openers, with a 1-2 record at Cowboys Stadium...The Redskins haven't won at Dallas since September 28, 2008 (26-24 at Texas Stadium in Irving)...The Cowboys are the first team in NFL history to play nine straight games decided by three points or fewer.