The two sides were level 0-0 after Friday's first leg and it took until the 69th minute of the second leg for the visitors to break the deadlock.
Zaha, who is joining Premier League champions United in July, sent Palace a step closer to joining him in the top flight with a header from close range after a superb cross from substitute Yannick Bolasie.
"I've got no words to express how I feel right now. It means the world to me, I've played for Palace for 10 years so doing this for Palace is just perfect for when I leave," Zaha told Sky Sports.
"I'm just happy I scored really. Getting two goals is just a dream really."
Brighton's best chances had come a few minutes before Zaha's opener with substitute Ashley Barnes having one effort saved on to the bar and another cleared off the line.
While Holloway was delighted with all his players, he was in no doubt about how vital 20-year-old Zaha's role had been in getting them to Wembley.
"I wish he was still with us next year," he said.
