Palace made the worst start to a top-flight English league campaign with no points or goals from their first seven matches, but they are now five points clear of the drop zone thanks to Bakary Sako's 21st-minute winner against the Clarets.
Hodgson's side have lost just one of their past 12 matches and are one point shy of the top half of the league.
"It changes quickly. You have to be very careful with statistics and I have no desire to start celebrating the fact that here, in the middle of January, we are in 12th place," Hodgson told a news conference.
"The only thing that interests me - and I have said it right from the start - is where we are on the final day of May and being one place above the relegation zone.
"Anything else apart from that would be a huge bonus, especially after being four games, four defeats, no points, no goals when I came (as a replacement for the sacked Frank de Boer). At that stage, things were looking very bleak.
"It's nice now that it no longer looks so bleak, and we should enjoy that, but there are going to be lots more games like the one today."
Sako scored the winner against Burnley by side-stepping defender Phil Bardsley and driving a powerful shot past goalkeeper Nick Pope for his seventh goal of the season.
The 29-year-old Mali international's contract expires later this year, but Hodgson is confident he will sign a new deal.
"Sako has come good for himself," the former England manager added. "Before this season he spent a lot of time on the bench and I don't quite know how many chances he has had, but he obviously hasn't done well enough in those chances to force himself into the team, but he has done very well recently for us.
"At the moment, it's pretty obvious that Bakary can sit back quite comfortably knowing full well that if he continues playing like he is playing at the moment, he is going to get a new contract, without any shadow of doubt."
(Reporting by Matt Westby; Editing by Christian Radnedge)
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