The German champions took a giant stride towards defending their title with the victory over the Bavarians and their Japanese midfielder was delighted
The win, which came thanks to Robert Lewandowski's 77th minute winner, moves Dortmund six points clear of Bayern at the top of the table and within reach of the title with four games to play.
Bayern had the chance to level the game late on, when Arjen Robben was brought down inside the penalty area, but his spotkick was saved by Roman Weidenfeller who had made the foul.
"It's pretty huge," Kagawa told reporters after the game. "For us to have won after Bayern got a penalty kick and Robben missed it is a huge blow to them. But from our perspective, we're really happy to have won a match like that."
Dortmund had the better of the first-half in the match but failed to create many clearcut opportunities.
"We played at home today so we were able to play our football. However, it's very disappointing that in today's first half we weren't able to score," Kagawa, who was replaced on 74 minutes, said.
The Japanese midfielder did seem to have more of a focus on defence for the game and he commented that that was the case as Bayern pressed in the second 45.
"If I dropped back, it made it harder for Bayern to speed up their attack. I was able to move and slide well, so the first half wasn't that scary," he said.
"But in the second half, our momentum decreased, and our balance wasn't so good so there were several dangerous moments. But despite that, we adjusted to their attack so we weren't that intimidated."
Dortmund's next match is away to derby rivals Schalke on Saturday, where they can edge closer to the title.
"The next match is becoming really important, so even though it's away we want to win," Kagawa added.