A drab first half saw Stuart Green fire Palace in front on 33 minutes, but Leicester recovered with second-half goals from Dudley Campbell and Patrick Kisnorbo.
However, Morrison made it 2-2 at the end with practically the last kick of the match.
An utterly miserable first 30 minutes eventually supplied some action.
Mark Kennedy slipped the ball into the path of Morrison only for the Palace hitman to be pulled back by defender Kisnorbo on the edge of the box.
Kisnorbo was booked for the offence but the Foxes were not going to get away with it.
From the free-kick, Green struck it low through a flimsy wall and into the far corner of the net.
Five minutes later Palace could have doubled their lead. Morrison was again in the action with a neat flick to Matt Lawrence, but the midfielder's shot from 12 yards out was superbly tipped over by the flying Marton Fulop.
Leicester didn't muster one attempt on the Palace goal in the first 45minuites, but it all changed after the break.
After 47 minuites Alan Sheehan fired over from Shaun Newton's cross and then midfielder James Wesolowski's effort was pushed over the bar by Julian Speroni for his first save.
The Foxes were on top and deservedly levelled on 64 minutes when skipper Stephen Clemence's inswinging corner was allowed to drop into the Palace goalmouth and Campbell forced it over the line as the home defence were unable to clear.
Manager Peter Taylor made substitutions to pick up his ailing side as Shefki Kuqi and Dougie Freedman joined the strikeforce.
The Eagles came close to restoring their lead when captain Mark Hudson had a free header which went straight into the arms of Fulop.
With three minutes left Leicester went in front with Clemence again providing the assist with a perfect corner.
Kisnorbo took full advantage of some awful marking with a low downward header past Speroni.
It looked all over for Palace until well into stoppage time when Freedman won a free-kick 20 yards out.
The veteran striker took the kick himself and, after Leon Cort got a touch, it fell straight to Morrison who made no mistake from ten yards out to save a point for Palace.