El Tri lost 2-0 in the last-16 contest, and Osorio was quick to apologise to the travelling fans afterwards.
"I would like to thank our fans and apologise that we didn't accomplish our goal and I'm sorry about that," he told reporters in his post-match press conference. "They are the definitely the best fans in the world.
"Being able to play at such a level with a team like Brazil, it shows Mexico is a good team. We just didn't have that extra quality they had in front of goal. We were able to play a balanced game and it says a lot about us and our style.
"We need to keep developing and we hope our players will come and play in Europe so they can play every weekend against the best clubs in the world, and then our national team will improve."