Morneau, the National League batting champion, drove a pitch from Japan starter Shintaro Fujinami over the wall in right to give his team a 4-1 lead in the third. It was the second homer of the series for Morneau, who also connected for a two-run homer in Game 2.
The win provided welcome relief for an MLB team that was no-hit by four Japanese pitchers in a 4-0 loss on Saturday.
"After getting no-hit yesterday, wanted to come out swinging the bats and we did," Morneau said. "We wanted to come here and represent MLB well. Unfortunately, we didn't do that in the first three games but I think we've gotten better."
Evan Longoria added to the lead with a solo homer off Japan reliever Minoru Iwata in the sixth.
After losing the first three games of the series and scoring only four runs, the MLB bats came to life in the first inning. Yasiel Puig doubled on a line drive to left and scored on Morneau's double to right.
Jose Altuve singled to right off Fujinami and Puig doubled to left in the second to set the stage for Morneau's blast that gave the MLB team all the momentum they needed.
"The top third of our lineup did an outstanding job," MLB manager John Farrell said. "I think eight out of our 10 hits came from Jose, Yasiel and Justin so that was a big part of our win."
Capuano picked up the win after holding Japan to one run on four hits while striking out four. Fujinami gave up four runs on five hits over four innings to take the loss.
"We dropped the first three games but we have a lot of pride and wanted to bounce back tonight," Capuano said. "The Japan hitters are very good so I was doing my best to keep them off balance and pitch smart."
Japan's only run came in the second inning when Nobuhiro Matsuda doubled off Capuano and scored on Yoshitomo Tsutsugo's double to center.
Japanese pitching sensation Shohei Otani will take the mound in Game 5 before his home fans at Sapporo Dome. Matt Shoemaker will start for the MLB team.