The Terriers defeated the Owls 4-3 on penalties after the second leg ended 1-1. They will now face Reading in the playoff final at Wembley Stadium on May 29.
"We are in the final; the fairy tale goes on and we want to write the next chapter," Wagner said.
"We were able to manage both legs. At the end, if you play two games and you have to go to the penalty shootout, it says how tight and even the games were.
"I am so proud and happy for our football club. They can celebrate tonight. Huddersfield Town will be at Wembley.
"Huddersfield Town is one game away from the Premier League. Even if I speak about it, it sounds ridiculous, but it is true."