"Everyone connected with the trip will be fairly compensated," baseball spokesman Rich Levin said.
Red Sox spokesman John Blake would not say how the dispute was resolved.
"We're going to Japan," he said.
Earlier, catcher Jason Varitek said the team would not take the field or go to Japan until Major League Baseball agreed to pay the coaches and staff.
Kevin Youkilis, the Red Sox player representative, said the agreement still must be put in writing and that the compensation for coaches and staff "is not the greatest thing that we wanted for them, but it's good."
Manager Terry Francona and his players became upset after learning staff members were not going to get a $40,000 stipend.
"We're so united. And I don't mean just the players," he said in a dugout interview with ESPN during Wednesday's game. "I mean the staff, the trainers and our players showed that, and that's what this was about. It wasn't about being greedy. It was about trying to be unified."
Daisuke Matsuzaka, who had been scheduled to pitch, left the stadium to pitch at a game against Minnesota's Triple A affiliate. David Aardsma started in his place. Matsuzaka is scheduled to be the opening day starter in Tokyo next week against Oakland.
Youkilis stressed the players felt strongly about not going to Japan without a resolution.
"The club's working on stuff and trying to get money where it needs to get," he said. "It was definitely an experience of a lifetime, and it ended in a good way."
Varitek said players thought it was necessary to take a stand on behalf of the coaches and staff.
"They're the basis of what takes care of us," he said.
At Oakland's spring-training site in Phoenix, the Athletics didn't take batting practice before their game against the Los Angeles Angels. Players met in the clubhouse while several got in contact with Red Sox players and the union.
A's player representative Huston Street emerged from the meeting and said the exhibition game would be played and Oakland players would make the trip.
"You have to stay firm in your belief, and I believe we've done that. Results have happened. That's why we're taking the field now. We wouldn't be taking the field now if we didn't firmly believe that the right thing was going to get done," he said. "The right thing is going to get done. We're going to play in Japan, and it's going to be an incredible series that everybody has been looking forward to."
A Boston player contacted Oakland pitcher Alan Embree on Wednesday morning.
"For those guys to take that stance - they're veterans. They feel strongly about it, and they brought it to the attention of higher-ups," Embree said. "We have to fix it one way or the other. ... Coaches deserved compensation. They're going over there, too, and every little bit counts."
Boston pitcher Curt Schilling said they learned Tuesday the deal was not what the players and coaches thought they'd agreed to with baseball.
"I think everyone was kind of caught off guard," he said.
Red Sox batting coach Dave Magadan said he appreciated the players' support.
"It means as much as the money itself," he told ESPN.
Oakland general manager Billy Beane was happy the trip will go on and expressed desire for additional international play.
"I hope we go to Rome. I hope we go to Paris, Berlin," Beane said, wearing shorts with a logo of the English soccer team Arsenal.
