It was the hosts who had the first effort of the match after seven minutes, but Scott Arfield's long-range tester was comfortably held by Julian Speroni.
Palace responded with a chance of their own just a few minutes later, but Alex Smithies parried away Yannick Bolasie's drive after the latter had cut in from the left.
The Terries came close just short of the half-hour mark when Neil Danns aggressively won the ball back before playing a one-two with Arfield, but he couldn't beat Speroni, who pushed the low effort behind.
The visiting keeper was called into action yet again just before the break when he saved from Danns, who volleyed his shot into the ground from a Calum Woods cross.
Andre Moritz had the next opportunity from a long-range free kick with 36 minutes gone, but his shot was saved low to Smithies' left.