Dortmund were coasting to victory after Kagawa had put them ahead in the 63rd minute, but goals in the last four minutes by Karim Haggui and Didier Konan stole all three points for Hannover.
The loss leaves Dortmund in 11th place in the Bundesliga standings, already eight points behind leaders Bayern Munich.
However, despite this latest setback, Kagawa still believes the club can bounce back and repeat last season's stunning title success.
"I am very disappointed to have missed the opportunity to win," Kagawa said in Sunday's post-match press conference. "But it is just a start of the season and we have enough chances to win [the league]."
Dortmund are still missing Lucas Barrios and Sven Bender through injury, while Sebastian Kehl and Mario Gotze were suspended for Sunday's match.
Kagawa believes his side will be boosted with the impending returns of these key players.
"Now we have to be patient. The injured players will come back soon."
Dortmund have only scored seven goals in their first five Bundesliga matches, but Kagawa does not believe the club have lost their ability to dominate games in the manner in which they did last term.
"Last year we could score more after our first goal [in each game]. We still have such momentum.
"We were confident when we got the opener, but we have conceded in such occasions [in the past few games].
"It means we have not shown enough strength throughout the games."
While he was happy at netting against Hannover, a feat he also managed last season, Kagawa admitted he would have been more pleased had Dortmund manage to come away with the win.
"I was happy for my goal, [but] I think I would be more happy if we could win."
Dortmund will now be looking to bounce back in their next Bundesliga fixture away to Mainz on Saturday.