Glentoran had the better of a first half which saw Larne trio Chris Gallagher, Mark Randall and Levi Ives all booked by referee Steven Gregg for hefty challenges.
Rohan Ferguson was called upon to deny Jordan Jenkins early on and then midfielder Palmer fired into the sidenetting after being set up by Chrissie Pattison.
Ferguson palmed away David Fisher’s strike, while at the other end Andrew Mills reacted quickly to smother Andy Ryan’s right-foot effort.
In the second half Frankie Hvid was unable to get on the end of a corner at the back post and Jenkins headed straight into the arms of Ferguson from close range.
After Lindsay’s thunderbolt opener, Ryan equalised from the spot with five minutes remaining and then almost grabbed a dramatic late winner when his effort flew past the post.
Larne had another opportunity late on when Mills excelled himself by palming away Joe Thomson’s piledriver.
In the shootout, Ryan, Joe Thomson, Chris Gallagher and Tomas Cosgrove were on target for Larne, while Jenkins, Lindsay, Palmer, Wassim and Chrissie Pattison were Glentoran’s successful penalty-takers.
