The Athletics will meet the Hanshin Tigers in another exhibition game Sunday before playing regular-season games against the Boston Red Sox on Tuesday and Wednesday night.
The Red Sox beat the Tigers 6-5 earlier Saturday in their first of four games in Japan.
The Giants took a 2-0 lead after three innings at the Tokyo Dome before Crosby and Hannahan pulled the A's even in the top of the fourth.
Crosby went to the opposite field with a homer over the right-field wall off Giants pitcher Takahiko Nomaguchi, and Hannahan tied the game with a 430-foot blast just below the scoreboard in center field.
"It was a fastball away," Crosby said. "They kept pitching in to me, so I got my foot down a little bit early into a fastball away and squared up pretty good."
It looked like the game would end in a tie but the A's scored two runs in the top of the ninth.
"Every time I get in the box, I just want to show what I can do," Bowen said. "Fortunately, I got a pitch I could hit."
Brown singled up the middle to score Bowen and make it 4-2. The Giants scored a run in the bottom of the ninth before Fernando Hernandez recorded the final out for the save.
Huston Street retired the side in the eighth inning for the win.
Oakland manager Bob Geren said his team was looking forward to playing the Red Sox.
"Our attitude is to try to win every time we step on the field, it doesn't matter who we are playing," Geren said. "I'm confident of what we can do."
Former Cleveland Indians outfielder Alex Ramirez gave Yomiuri a 2-0 lead in the bottom of the third with a single to right off Oakland starter Dana Eveland, who gave up two runs on seven hits in five innings.
The Giants went ahead 1-0 in the second inning on a Ken Kato RBI single off Eveland.