Goals from substitutes Shinji Okazaki and Keisuke Honda gave Japan a 2-0 win over Tunisia in Oita on Friday.
It marked an ideal start for Halilhodzic, who replaced Javier Aguirre after the Asian Cup.
“He's a good fit with us, I think," Kagawa told the Japan News.
"He demands things intensely, and strength and speed are qualities that Japan has lacked.
"There are still issues we have to work on, but everyone has an awareness of the kind of soccer the coach wants - precise rotation of the ball, quick build up, and to switch well between defence and attack."
Halilhodzic made several changes to Japan's starting XI for the clash against Tunisia, but he may add class to his team with Honda and Kagawa from the outset.
Only one of those – the last clash between the two in Toyota in 2012 – has gone Uzbekistan's way.
Uzbekistan's first clash since the Asian Cup was also on Friday, when they were held to a 1-1 draw by South Korea.
Zokhir Kuziboyev scored Uzbekistan's equaliser after they had fallen behind in Daejeon.
