Palace were unlucky to be 2-0 down at the break, having created a number of excellent chances, only to be punished when Alexander Isak and Andy Robertson converted the Reds' first two shots on target of the game.
But the Eagles refused to lie down and got themselves back into the game when Daniel Munoz scored from outside the box with just under 20 minutes remaining.
Jørgen Strand Larsen was then hugely unlucky to hit the post as Palace piled forward, sensing a draw was there for the taking.
Ultimately, Liverpool got the vital fourth goal to make it 3-1 in the dying seconds through Florian Wirtz's powerful effort from inside the box.
For Glasner the game was decided by the difference in finishing between the two teams.
"The difference was the efficiency," the Palace Manager told Sky Sports. "Liverpool were clinical and we were not.