Griffey followed Jay Bruce's one-out double in the first with a 344-foot line drive that bounced off an advertising sign over the bench in the visitor's bullpen down the right-field line.
"See it and hit it," Griffey said about his approach to facing a pitcher for the first time. "I was trying to get a pitch to pull and try to get Jay to third, one step closer to home. I just happened to hit a line drive that went out. The most important thing was to get the guy over."
It was Griffey's 11th of the season and career homer No. 604, leaving him five shy of tying Sammy Sosa for fifth on the all-time list. It also gave the nine-time All-Star 5,001 total bases, becoming 18th player to reach that level.
"It was a fastball," said Bergmann, who became the 387th pitcher to surrender a home run to Griffey. "I was trying to go in on him and left it out over the plate. He's hit 604 homers. You wish it didn't happen, but it gives me incentive to do a little better next time."
Arroyo (6-7) was sharp after the game started following a 1-hour, 47-minute rain delay. He gave up five hits to earn his second consecutive win after three straight losses.
"This is the best I've felt during a day game," said Arroyo, whose last start was a day game and whose next outing in Chicago also is scheduled to start during the day. "I think the start being pushed back a couple of hours helped. It gave me a chance to wake up and feel a little more normal."
Washington, which has been shut out 10 times this season, had runners at first and second with one out in the ninth but Cordero got Willie Harris to fly out to right and struck out Cristian Guzman to end the game.
Fireworks went off prematurely when catcher Paul Bako couldn't hang on to Guzman's two-strike foul tip, but Guzman took a called strike on Cordero's next pitch.
"I was hoping that wasn't a bad omen," Reds manager Dusty Baker said. "I think that's the first time I've seen that. They need to wait and make sure before they press the button."
Cincinnati went ahead 3-0 in the fourth. Bako walked with two outs and scored when left fielder Elijah Dukes misplayed Arroyo's line drive for a three-base error.
Bergmann (1-6) allowed three runs, two earned, and six hits in six innings. He is 0-5 in nine starts since he beat the New York Mets on May 15.
"Regardless of the home run, he went six innings and did a tremendous job," Nationals manager Manny Acta said.
The Nationals, who had at least one baserunner in all but one inning against Arroyo, have scored a total of eight runs while Bergmann was in the game over his last nine starts.
Baker said OF Norris Hopper would go on the disabled list Saturday and undergo season-ending surgery on his elbow, which had bothered him all season and kept him on the DL for two months. ... RHP Josh Fogg, on the DL since June 1 with back problems, will be activated and start in place of RHP Aaron Harang on Saturday against Washington. Harang will start Tuesday against the Cubs at Chicago ... Guzman singled in the third inning to extend his hitting streak to 12 games, matching the season high he set June 7-19. ... It was Cincinnati's fifth shutout.