Morrison put the Eagles ahead when he volleyed home from Gary Borrowdale's hopeful ball into the penalty area.
Dougie Freedman fed McAnuff to slot home Palace's second but Paul Furlong's header brought QPR back into the game.
Richard Langley had already gone close twice for Rangers and only late saves by Gabor Kiraly stopped Lee Cook and Marc Nygaard levelling late on.
QPR caretaker manager Gary Waddock:
"Letting in that goal in the third minute meant our gameplan went out of the window.
"But we were really positive in the second half, and gave them a run for their money."
Crystal Palace: Kiraly, Ward, Borrowdale, Boyce, Leigertwood, Watson, Soares, Andrews (Hughes 78), McAnuff, Macken (Freedman 15), Morrison.
Subs Not Used: Speroni, Fray, Grabban.
Booked: Leigertwood.
Goals: Morrison 3, McAnuff 32.
QPR: Jones, Bignot (Furlong 37), Milanese, Evatt, Shittu, Cook, Donnelly (Baidoo 71), Lomas, Langley, Rose, Nygaard.
Subs Not Used: Santos, Kus, Thomas.
Booked: Milanese, Nygaard.
Goals: Furlong 56.
Att: 17,550
Ref: A Hall (W Midlands).