Heading into the match with a 3-2 first leg advantage, Kashiwa found themselves trailing 2-0 when Jong Tae-se and Ku Ja-ryong scored in the 26th and 54th minute respectively.
Kashiwa bounced back from the double strike when Yusuke Kobayashi scored a crucial goal in the 65th minute.
Even though Kashiwa lost the match 2-1, they advanced to the last eight on the away goals rule.
"We got through over the two games and I am both happy and relieved," Yoshida told the Japan Times. "We knew we would be under pressure for 90 minutes but we coped by making the adjustments that we needed to make through the game.
"We withstood their counterattacks and managed to get through by playing tenaciously. The players worked hard, ran hard and fought hard so all the credit to them.
"Things have been up and down for us in the J. League but we are competing hard in the ACL. Advancing in the tournament can give the players something extra, a belief that we can keep going further. That is the power of this tournament."
Yoshida admitted that progressing to the quarter-finals wouldn't have been possible if his side hadn't scored three away goals.
"Getting three away goals gave us a bit of an advantage but we wanted to forget the away result and just try to win this game," he said. "We conceded twice but got the result because we refused to give up and it’s great that we have got to the quarterfinals."