The Asian champions were considered a chance to make an impact in Brazil, but they collected just one point to finish bottom of Group C.
Alberto Zaccheroni resigned from his position as coach in the wake of Japan's 4-1 loss to Colombia on Tuesday.
Kagawa said there was a lack of quality in the Japan squad, meaning they were unable to produce in the big moments.
"We can play well as a team, but in terms of individual quality, we are not good enough to score or avoid conceding in the final moments, which is the difference between us and the best teams in the world," the Manchester United man said.
"We actually conceded when we had to try not to, and couldn't finish it when we needed to score against Greece. I think that made a very big difference.
"I would like to get it right so as to improve."
Asked about winning a place in the United starting line-up under Louis van Gaal, Kagawa said he was yet to think about it as he recovers from the disappointment of the World Cup.
"Firstly I have to reflect on this World Cup," he said.
"This is not the end of my career, so I want to make the best of this frustration and disappointment for the sake of my future career."
Japan defender Yuto Nagatomo, 27, was inconsolable as he spoke about his team's performance.
"Honestly I couldn't sleep yesterday and was thinking of things such as my future international career and what I have to do now, but I have given everything for this World Cup," he said as tears filled his eyes.
