"I'm delighted we're going to Wembley. Tonight another little bit of history has been made for this football club. It's terrific for everyone here," the Bluebirds boss said.
"Anyone who's watched our games this season can see that our players just don't know how to lie down and die. I had great belief in our determination and spirit to keep going."
Mackay also praised Cardiff City goalkeeper Tom Heaton, who denied two Palace kicks in the shoot-out.
"Tom has played second fiddle to David Marshall in the league this season, but he's been marvellous right through the Carling Cup," he said.
"He took a bit of criticism after their goal in the first leg, so I'm delighted for him."
Cardiff will face either Manchester City or Liverpool in the Carling Cup final.