Palace were up against it in the first half, with Walter Benitez making four superb stops to go into the break goalless.
The Eagles had a new lease of life after the interval and went toe-to-toe with the Premier League leaders in an entertaining half of football.
Palace looked to be heading out when Maxence Lacroix turned the ball into the back of his own net in the 80th minute following a hectic scramble from an Arsenal corner.
But Palace had other ideas and kept fighting. A stunning free-kick from Adam Wharton was turned into the Arsenal net by Marc Guehi to force penalties in the fifth of nine additional minutes.
After 15 successful spot-kicks, Lacroix stepped up but saw his effort saved by Kepa Arrizabalaga, with Palace suffering Carabao Cup quarter-final defeat against Arsenal in successive seasons.
Maxence Lacroix holds hands up to Crystal Palace supporters after missing decisive penalty against Arsenal (Image: Nathaniel Hutchinson)
Glasner’s side came into the match after three games without a win, having conceded four set-piece goals away at Leeds United on Sunday in a 4-1 loss.
Palace boss Glasner, however, was impressed by the spirit his side showed on Tuesday.
“When you lose a game on penalties, you're always disappointed,” said Glasner.
“On the other hand, I'm delighted with the second half performance today, especially after the really bad first half performance.
“Walter Benitez kept us in the game. The second half was much better, it looked like an even game.
“We’ve already had two penalty shootouts this year, we won both – today Arsenal were the lucky team.
“I've watched many penalty shootouts in my life, but I can't remember having seen so many well-taken penalties.
"Somebody always has to miss, and today it was one of our players, but nobody to blame.
“It's important to build on the second half performance and also show the great character again.
“In the second half, 1-0 down from a set play, it looked like deja vu from the Leeds game.
“Then, coming back with everything that happened in the last weeks and months is really a huge credit to the players for their mindset.
“The way we switched the performance should give us confidence. We’ll take many positives from today’s performance.”
Crystal Palace players line-up for penalty shootout (Image: Nathaniel Hutchinson)
Glasner has given his players the next two days off after playing three games in six days, allowing them to enjoy the festive period, with no Boxing Day fixture scheduled this season.
The Palace coach urged his squad to switch off, with the Austrian heading home to celebrate Christmas with his family before the players report back to the training ground on Boxing Day ahead of facing Tottenham Hotspur on December 28th.
“The next two days, the players don't hear anything. It's so important to switch off and not talk about football.
“Our players have families, so the parents deserve for their son to come home and not the football player.
“The wives, the girlfriends, the children. The dad and the husband are now at home – not the football player.
"Don’t talk about football for two and a half days, really switch off completely because then we have five games in the next two weeks.
“It’s really about leaving football, then when we meet on the 26th in the afternoon, we will analyse this game today.”