Timothy Fosu-Mensah of Crystal Paalce is challenged by Ashley Barnes of Burnley Photo: Getty
But Bakary Sako's first-half finish at Selhurst Park instead took them up to 12th and five points clear of the bottom three, and at a time when they have eight first-team players absent through injury.
Hodgson said: "It changes quickly. It became seven games, seven defeats. At that stage things were looking very, very bleak. It's nice now that things don't look as bleak. We should enjoy that but there will be many more games like (against Burnley) where you do enough, especially in the first half, to merit the three points but then you suffer in the second.
"I am very careful about statistics and I have no desire to start celebrating here in the middle of January. I said when I came here that the only thing that matters to me is where we are on the final day in May. The only thing that interests me is being one place above the relegation zone. Anything more than that would be a huge bonus."
Clarets manager Sean Dyche said: "It's the simplicity of Palace's play. We didn't really deal with that.