Palace have the longest unbeaten home run in the Championship with 13 matches, having won eight and drawn five since losing their Selhurst Park opener against Watford on 18 August.
For the visitors, Chris Eagles was in good form, twice forcing Julian Speroni into saves before Palace thought they should have had a penalty.
The ball appeared to hit the arm of Tyrone Mears but referee Robert Madley ignored the appeals.
The home side also thought they had opened the scoring at the end of the first half. Glenn Murray reacted quickest after Adam Bogdan saved Yannick Bolasie's shot but the linesman was flagging for offside.
Another jinking run and cross from Zaha should have produced the winner for Murray, but the Championship's top scorer, with 22 league goals, scooped his shot over the bar.
At the other end, Davies headed straight at Speroni before the Argentine saved well from Keith Andrews.
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