Having earlier saved Steven Naismith's penalty, Adrian had to watch as team-mate Stewart Downing failed to convert past Everton goalkeeper Joel Robles - pushing the shoot-out to sudden death.
After nine penalty kicks each, Robles stepped up for Everton but blasted his effort onto the bar and the two goalkeepers switched places.
"At that moment I feel so quiet, so relaxed. I had no nerves," he told West Ham TV after the game.
"And I took off my gloves and I thought 'this game is going to end' and I strike."
Having sent West Ham into the FA Cup fourth round, Adrian added: "I feel really happy. Not just me but for all my team-mates, for the fans, for the club, because it's a good moment for us and we keep going in the FA Cup."
Roberto Martinez's side could not hold on, however, with Carlton Cole making it 2-2 in the 113th minute, and with both teams wasting top chances to win the match before the shoot-out, penalties were required.
"It's a crazy game for us," Adrian said.
"We played against 10 players and when they scored the second goal it was really hard for us. But the team kept going and going."
