Marco Silva's substitutions made the difference for Everton as they struck late in a 2-0 win over Palace, who had a penalty saved by Jordan Pickford with the game at 0-0.
Palace remain two points above the drop and face Arsenal, Chelsea, Tottenham and Manchester United in their next four league games.
Hodgson has felt defeat in six of his last eight games in charge and is sick of the sinking feeling.
"The first goal was a massive blow after defending so well," he said to 'Sky Sports'.
"Also, the chance to put ourselves in a commanding position and not take it from the penalty.
"It's hard to stand here as a loser again - I could have been standing here as a winner or someone who has coached their team to a positive point but they scored the goals and we didn't.
"The fact is you have to take your chances and we came across a goalkeeper in form who made two outstanding saves in the first half which meant we came in at half-time 0-0, and we had the penalty which would have made a difference.
"It definitely lifted the crowd and they really got behind the team and gave them fresh impetus going forward.
"Luka has only missed two penalties out of around nine or 10 so I'm sure he'll score the next one."