A first-half effort from Bakary Sako was all that separated the two sides at Selhurst Park, in the process condemning Sean Dyche's men to back-to-back defeats and leaving them without a win in six after an impressive first half of the campaign.
For the Eagles, meanwhile, the result represented back-to-back victories and lifted them up to 12th in the Premier League table, five points clear of the dropzone.
"It was a massive three points. We knew it would be very, very tough. In the first half we created chances to make life more comfortable for us in the second half," Hodgson told reporters afterwards.
"We knew they would get stronger and more questions would be asked of our back four and goalkeeper. I'm delighted we were able to withstand that test. They are very good at what they do, a strong team, a physical team.
"At least now we can look at the league table for a week or two without shivers running down our spine.
"We are [still] looking to survive. Every three points is three points closer. Everyone at this club, fans, directors, coaches, we will be delighted if at the end of the season we are a point above the relegation zone. We will never waiver from that. Today the three points was everything and we got them."
Next up for Palace is a difficult trip to Arsenal next weekend.