After SoftBank won the first two games in Yokohama, the BayStars were desperate for a win to avoid a possible sweep in Game 4 on Wednesday.
Kuwahara gave his team a 1-0 lead in the first inning at Mizuho PayPay Dome when he doubled off SoftBank starter Carter Stewart Jr, advanced to third on a sacrifice bunt and then scored on a groundout.
"Kuwahara set the pace for us in this game," said Yokohama manager Daisuke Miura. "It was a game we really needed to win."
But SoftBank quickly got the run back in the bottom of the frame on a double by Kensuke Kondoh that scored Yuki Yanagita from second.
After four innings, SoftBank manager Hiroki Kokubo took out Stewart, who gave up one run on three hits while striking out four and walking three batters.
It was a move Kokubo may have regretted.
In the fifth inning, Kuwahara greeted SoftBank reliever Ryosuke Ohtsu with a solo homer to left on the very first pitch he saw to make it 2-1.
Ohtsu then loaded the bases and Yoshitomo Tsutsugo just missed a grand slam by inches but drove in another run on a sacrifice fly snared up against the wall in right by Yanagita.
Kokubo quickly turned the ball over to Shunsuke Iwai, who also loaded the bases but got out of the jam with no further damage.
BayStars Take a 4-1 Lead in the 8th
Yokohama catcher Yasutaka Tobashira doubled off reliever Kazuki Sugiyama in the eighth to score pinch runner Takuma Hayashi from second and make it 4-1.
Left-hander Azuma, who went 13-4 in the regular season but was injured during the second stage of the Central League Climax Series, scattered 10 hits. He struck out four and gave up one run to pick up the win in his first mound appearance since October 12.
"I was absent during the Climax Series so I just wanted to go out and do my best," Azuma said. "I know they have a strong lineup so I just tried to keep them off balance."
He added, "I didn't think I could go seven but was happy to contribute to the win."
Relievers Ise, Morihara Finish Strong for the BayStars
Hiromu Ise pitched a scoreless eighth and Kohei Morihara retired the side in the ninth for the save.
Ohtsu was the losing pitcher after giving up two runs on two hits and recording just one out in the fifth.
Yanagita, a 36-year-old veteran, went 3-for-5 with one run scored. He also made a superb leaping catch of Tsutsugo's blast in the fifth.
The BayStars, who eliminated the defending Japan Series champion Hanshin Tigers and the Central League champion Yomiuri Giants are bidding to become the second team in NPB to finish third in the standings and win the Japan Series.
In 2010, the Chiba Lotte Marines achieved the feat when they beat the Seibu Lions and SoftBank Hawks in the Pacific League Climax Series and then won the Japan Series by beating the Chunichi Dragons with four wins, two losses and one tie.
The Fukuoka-based Hawks have won six of the last 10 Japan Series, including in 2017 when they beat the BayStars four games to two. Their 14-game Japan Series winning streak dating to 2018 ended on Tuesday night.
A Look Ahead
Shuta Ishikawa is expected to start Game 4 for the Hawks on Wednesday, while Anthony Kay is scheduled to go for the BayStars.
Game 5 is set for Thursday, also at Mizuho PayPay Dome.