Substitute Justin Devenny scored the winning spot kick to give Palace the spoils in the traditional season curtain-raiser on Sunday after Mohamed Salah ballooned the Reds' first penalty and Henderson then saved efforts from Alexis Mac Allister and Harvey Elliott.
"I love it, I love the big moments, I love being in that pressure moment," said Henderson, who also saved an Omar Marmoush penalty as Palace won the FA Cup, their first major trophy, against Manchester City in May.
New recruit Hugo Ekitike had put Liverpool ahead inside four minutes after he and record signing Florian Wirtz combined beautifully before the Frenchman produced a superb finish.
But Palace deservedly levelled in the 17th minute, Jean-Philippe Mateta sending Alisson the wrong way from the penalty spot after Virgil van Dijk brought down Ismaila Sarr.
Close-season signing Jeremie Frimpong restored the Reds' lead, as Liverpool fans marked the death of Portugal forward Diogo Jota with 20 minutes on the clock in honour of his shirt number, sending an attempted cross looping in off the post.
The Eagles posed a threat on the counter and kept themselves in the game, although Ekitike spurned two decent chances to extend Liverpool's lead in the early stages of the second half.
Palace's England forward Eberechi Eze then tested Alisson with a low shot before the Londoners equalised again with 13 minutes to play, Sarr slotting past the onrushing Alisson.
Palace nearly grabbed a winner at the end, with Mac Allister perhaps fortunate to have a VAR referral for handball in the box go his way before Devenny nearly scored in added time, as the traditional match between the league champions and the Cup holders went to penalties.
Devenny got the winner in the shootout with an emphatic penalty to give Palace a third victory at Wembley in a row.
Palace's celebrations last season soon turned into a court battle for the club after they were demoted from the second-tier Europa League to the third-tier UEFA Conference League.
European soccer's governing body found Palace in breach of multi-club ownership regulations because of their ties to French club Lyon through American businessman John Textor.
Palace lodged an appeal with the Court of Arbitration for Sport with a verdict expected on Monday.
Before the match started, a tribute to former Liverpool star Diogo Jota and his brother Andre Silva, who died in a car accident last month, was marred by some fans as players stood for a moment's silence before kickoff.
Loud voices could be heard in the crowd, which prompted some boos, and referee Chris Kavanagh then blew his whistle to signal the end of the tribute.
Liverpool have retired Jota's No. 20 jersey as a mark of respect, with fans applauding as the clock struck 20 minutes during the game.
