Mussina (10-5) allowed four runs and 10 hits in eight innings, dropping to 0-2 in two career starts against Cincinnati. It was his first loss since he recorded just two outs against Baltimore on May 20.
New York hosted Cincinnati for the first time since the Reds won 7-2 on Oct. 21, 1976, to complete a four-game World Series sweep - the third and last time the Yankees were swept in the Fall Classic.
Griffey's father scored the winning run in the ninth inning of the Reds' 4-3 win in Game 2.
The two franchises have met in the World Series three times, with the Yankees winning in four games in 1939 and five games in 1961.
Cincinnati took two of three from the Yankees at Great American Ball Park in 2003 but had never played a regular-season game in the House that Ruth Built before Friday.
Griffey, the longest-tenured Reds player, downplayed the significance of the matchup before the game but a sellout crowd of 53,421 gave the newest member of the 600-home run club a warm reception before each of his at-bats. Cameras also flashed in the stands with Griffey at the plate.
Griffey grounded an RBI single into right field to finish Cincinnati's three-run fifth against Mussina, who was 9-1 with a 3.25 ERA in his previous 11 starts. The Reds' run started when leadoff hitter Edwin Encarnacion's sinking liner went off left fielder Johnny Damon's glove for a double.
Cabrera had the big blow of the inning, hitting a tiebreaking two-run double with two outs to give the Reds a 3-1 lead. Cabrera's drive into the left-field corner came after Mussina issued an intentional walk to Jay Bruce to face the light-hitting shortstop, who entered with a .214 average and no RBIs.
Damon had a broken-bat RBI single in the third and Robinson Cano's base hit drove in another run in the seventh but that was it against Volquez (10-2), who entered leading the majors with a 1.64 ERA and 105 strikeouts.
The Yankees put runners on first and third after Cano's hit cut it to 4-2 but Volquez struck out Melky Cabrera to get out of the inning, pumping his arm as he made his way to the dugout. The hard-throwing right-hander could make another start at Yankee Stadium next month - in the All-Star game.
Jared Burton pitched the eighth and Francisco Cordero finished for his 14th save in 17 opportunities. Hideki Matsui singled with two outs in the ninth but Jason Giambi grounded out to end the game.
Jolbert Cabrera singled in the seventh and dislocated his left index finger sliding into second after Mussina threw a wild pitch.
He was taken to a hospital to have it reduced under anesthesia.
Paul Janish came in to run for Cabrera and replaced him in the field in the bottom half. The Reds already have three shortstops on the disabled list.
