With the tie level at 2-2 and extra-time looming, referee Michael Oliver pointed to the spot after Marco Cassetti nudged Anthony Knockaert to the ground.
Knockaert stepped up to take the penalty which would have sent Leicester to Wembley but Almunia saved it, and the rebound, before Watford broke away and Troy Deeney netted the winner in the 97th minute.
"It was a dramatic one, the kind of game you dream about and sometimes it happens," said Almunia.
"It's the good side of football but the bad side is for Leicester, I feel sorry for them.
"When the ref gave the penalty I was nearly thinking of my holidays, there were only a couple of minutes left.
"But I was very relaxed. We watched the penalty takers on a DVD but this guy wasn't on it. So I did nothing special, I just guessed the right side and it hit my legs.
"The ball dropped in front of me, I saw the guy come in and the ball hit me again. Then I thought football might give us something good in the last minute and Troy finished the job.
"It was great in the dressing room afterwards; singing, jumping around to music and some beers - just to rehydrate the body."
Former Arsenal keeper Almunia will captain the Hornets at Wembley in a fortnight.
- Sports & Recreation
- Troy Deeney
- Anthony Knockaert