Clear-cut chances were few and far between for two sides who showed little cutting edge throughout the 90 minutes, as Steve Cooper's side stretched their winless top-flight run to four matches.
Both sides were deprived of their key attacking threats heading into the evening showdown, as Jairo Riedewald came in for the Eberechi Eze-less Palace, while Chris Wood led the line for Forest in the absence of Taiwo Awoniyi.
Hodgson's men did not allow Forest a sniff in the earliest exchanges, but the Eagles lacked a killer instinct up front, and the first opening of note came the visitors' way with a quarter of the game gone.
A long ball over the top from Murillo found the boot of Morgan Gibbs-White, who went for a chipped effort with Sam Johnstone in no man's land, but the ball bounced off the post and into the grateful palms of the Eagles goalkeeper.
Palace's injury problems continued to pile up, as Jeffrey Schlupp came off with an issue in the 26th minute to be replaced by Jesurun Rak-Sakyi, although the hosts managed to carve out an opportunity in the 33rd minute, as Jordan Ayew tried to volley home from a Joachim Andersen cross, but the ball hit his shin and sailed into the stands.
A low-key end to the first half was seemingly in store, but the impressive Murillo almost conjured up a moment of magic in added time, embarking on a wonderful solo run from just inside the Palace half and only being denied by a strong hand from Johnstone, who had a more comfortable save to make from his follow-up volley.
A half-time change from the Tricky Trees saw Gonzalo Montiel replace Serge Aurier - who had sustained an injury midway through the first half - and there was very little to write home about for either side in a drab opening to the second period.
Palace attempted to make a few inroads without success, and a rare foray forward from Forest saw Montiel send a half-volley towards goal in the 60th minute, but Marc Guehi was in the way.
Three minutes later, Jean-Philippe Mateta spurned a magnificent chance for Palace, letting the ball roll across him just inside the box and slipping at the crucial moment, sending his low strike just wide of the goal.
Matt Turner eventually had a couple of saves to make, comfortably denying Mateta and Ayew in quick succession, before Andersen came up with a crucial block to keep out Montiel's Paolo Di Canio-esque scissor kick volley in the 78th minute.
The hosts suddenly had their backs against the wall as the game drew to a close, and Nicolas Dominguez attempted to pick out the far corner from the edge of the box in the 89th minute, only to be denied by a strong full-length save from Johnstone.
Following the international break, ninth-placed Palace find themselves back in action against Newcastle United at St James' Park on October 21, while Forest - who drop to 13th - host Luton Town in 14 days' time.