Palace are without a goal in their opening five league games, a Premier League record, but not for want of trying with Ruben Loftus-Cheek twice going close after half an hour and Forster also saving from Christian Benteke.
Hodgson took charge at Palace this week after Dutchman Frank de Boer was sacked following four defeats and no goals in his debut Premier League season and faces a tough challenge to rebuild confidence and turn the team's form around.
"I still believe in the long-term, these players will pick themselves up from where they have been knocked down to," he said. "It's going to be a long road and there might be some more pain before we see some light at the end of the tunnel."
Palace’s long wait for a first league goal of the season could have ended 10 minutes after Saints went ahead through Davis when Benteke met Loftus-Cheek’s low cross at the near post but Forster got his body in the way to make the save.
Palace’s impressive midfielder Loftus-Cheek came close when he skimmed the outside of the post with a shot from the edge of the box then had an effort deflected for a corner.
Two more chances went begging in a downpour early in the second half when another Loftus-Cheek centre flew across the face of goal before Forster saved from Jason Puncheon.
There were boos from the Palace crowd when Hodgson brought off Loftus-Cheek, his best player, for Bakary Sako with 15 minutes to go and the change did not work for his goal-shy team.
Saints, who had last scored when they beat West Ham United 3-2 at home on Aug. 19, notched their second win to move up to provisional fifth spot on eight points before the afternoon kickoffs, while Palace are rooted to the bottom of the table.
"You have to fight for every single point. You could see today that Crystal Palace were good," said Southampton manager Mauricio Pellegrino.
"They created chances in the first half and pressed higher in the second but in general we controlled the game. We could have scored another goal but playing here is really difficult."
(Reporting by Rex Gowar; Editing by Ken Ferris)
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