The Black Cats booked their place at Wembley Stadium by defeating United on penalties, with the aggregate scoreline tied at 3-3 following a 2-1 loss at Old Trafford.
"It has been a remarkable achievement from the lads and they've dug in the last three months and worked their socks off and you know we are getting the rewards from winning at places like this," Bardsley told Sky Sports News.
Ex-United man Bardsley appeared to have knocked his former club out of the competition in extra time when his tame effort fortuitously slipped through the hands of Red Devils goalkeeper David de Gea, but Javier Hernandez's dramatic late strike took it to penalties.
"Soon as I hit the target I thought you've always got a chance," Bardsley said of his goal. "I think it was a little bit lucky it wobbled a little bit in front of De Gea and he threw it in a little bit, but they all count."
Sunderland will take on Manchester City in the League Cup final.