The tie, which finished 3-3 on aggregate, was a rematch of last season's Champions League final which Bayern won 1-0. Prior to Tuesday's game, Bayern's Kimmich and Thomas Muller said they were confident the German champions would go through at the Parc des Princes.
But it was PSG who advanced, with Neymar celebrating with teammate Leandro Paredes in front of an annoyed Kimmich as the referee whistled for full-time.
"It's funny because I didn't even celebrate to mess with him, it was more with Leo and I ended up celebrating with him. It was fate that put me close to [Kimmich]," Neymar told TNT Sports Brasil after the match. "He said that their team was better, that they would win. He was sure that they would go to the semifinals."
Bayern won 1-0 despite suffering on the break throughout the match, with Neymar hitting the woodwork twice, but it was not enough for them to progress.
However, that did not stop Bayern coach Hansi Flick from saying his side was more deserving to move on.
"Over both games I think we were the better team," said Flick, who had to cope without injured striker Robert Lewandowski for both games. "We've had some back luck, we were missing some players but we have to accept it."
Responding to Flick, PSG coach Mauricio Pochettino said the German players' pre-game comments were fair.
"It is normal that they were confident, they're a team who have won everything and they thought they were the favourites," said Pochettino.
"Football is such a beautiful game but it's also a game in which you never give up and you have to fight. But I understand why they said they were confident, it's normal to play mind games," he added.
"The message we had to send was given last season, when we reached the final, played a great game against Bayern, which was the best team last year. This year I didn't see it that way," Neymar said. "During the draw, I told my teammates that we were going to face Bayern, and it happened.
"I told them to be calm, that we were prepared. We needed to show this to ourselves, that we can do it. It was a great game, even though we were beaten, we played better."
The club and Neymar also in the middle of a contract negotiation, but the ex-Barcelona man said he wants to stay in Paris.
"I don't think this is even a topic anymore, I obviously feel very comfortable, at home at PSG,'' Neymar said. "I feel happier than before.''
The Associated Press and Reuters contributed to his report.