A flurry of chances ensued from the restart, with one passage of play seeing Morrison denied by Lee at one end and, seconds later, Pecsi tipping wide a Dean Benamar strike after a surging run.
Amara Nallo sent a back-post header off target from a corner kick, while Lucas Pitt timed a sliding block superbly to prevent Marsh finishing a turn inside the area.
The next goal was going to be pivotal – and Liverpool got it.
Two minutes shy of the hour mark, Morrison’s dribbling on the right wing created room for a teasing low cross that Wright pounced upon to clinically tuck in.
Back came Palace immediately and Marsh was able to line up a strike from a central position 10 yards out that rattled away off the crossbar.
With Morrison a constant spark, the Reds were threatening a third, but Pecsi had to correctly judge a smother when Ben Casey snuck into a one-v-one.
The end of normal time arrived with the score still 2-2 and an additional half-hour began with Liverpool taking the upper hand.
Yet another Morrison scamper into the box forced a clearance off the goalline, although as fatigue steadily set in, the chances dried up.
A lovely Morrison delivery just evaded the arriving Josh Sonni-Lambie yards out, before, on 107 minutes, the Eagles regained the advantage.
The ball was not cleared from a deep free-kick lofted into the area and eventually Farquhar controlled it and prodded a finish past Pecsi.
Reds substitute Figueroa immediately had a near-post nod deflected wide after Morrison’s latest run and pinpoint cross.
But the USA youth international did not pass up a second chance, meeting Calvin Ramsay’s hoisted cross from the left side with a brilliant downward header to make it 3-3.
No further goals meant it went to penalties, with Lee keeping out the efforts from Michael Laffey, Wright and Figueroa to send Palace through.
Liverpool: Pecsi, Ramsay, Nallo, Ndiaye, Pitt, Kelly (Laffey, 60), Morrison, McConnell (Pilling, 60), W. Wright, Nyoni (Bradshaw, 91), Sonni-Lambie (Figueroa, 106).
Unused sub: Misciur.