The presentation was made by students from Showa Boston, which is an institute for language and culture.
Senbazuru is a colorful display that is made of 1,000 paper cranes that are tied together. They are used in Japan for several purposes, one of which is to wish sports teams well.
"Daisuke and I feel a responsibility to do well for these students who spent so much time making these," Okajima told Japanese reporters.
It is not uncommon for teams to have Senbazuru hanging from the dugout in the Koshien High School tournament, which is a big event in Japan.
Three students, wearing kimono robes, made the presentation. Matsuzaka and Okajima both nodded several times in appreciation.
"We have many high school students come to our camp for two-week programs," said Kenji Yuasa, assistant director of student services at Showa Boston. "We wanted our students to be able to do something to teach about Japanese culture, and we came up with Senbazuru, which is hopes and dreams.
"With Matsuzaka and Okajima on the Red Sox, we really want them to be great. [The students] have made 1,000 cranes. We have two sets."
Yuasa was overjoyed that the Red Sox allowed the school to come to the ballpark to make their presentation.
"We wanted to present it to the Red Sox. We had no idea we'd be able to," Yuasa said. "So we just made it and were hoping and dreaming to be able to bring it over, and it came true."
Because of obvious space limitations, the Red Sox won't be able to put the Senbazuru in the dugout. Okajima said he might display his at his Boston home.
Ian Browne is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.