The Eagles are the first team in English top-flight history to open a season with five consecutive defeats without scoring as the Saints conducted a smash and grad raid at Selhurst Park.
Hodgson was quickly made aware of the size of the task facing him as Palace boss when Davis scored the only goal of the game after six minutes.
The visitors were able to take advantage of some woeful Palace defending, Dusan Tadic getting in down the right and delivering a low cross which Wayne Hennessey palmed away, only for Davis to slot the ball in to an unguarded net having reacted quickest.
Palace came close to hitting straight back though, Southampton goalkeeper Fraser Forster making a great save to keep them ahead, denying Christian Benteke at point-blank range after a good run by Ruben Loftus-Cheek.
The lively Loftus-Cheek then worked the one two with Benteke before slamming his long-range shot narrowly wide.
Forster was drawn in to the action again soon after half-time, the England stopper making another outstanding save from close range to deny Jason Puncheon after Jeffery Schlupp was given too much space to get his cross in.
The Saints keeper was badly under worked from that point onwards as Southampton threatened to grab a second on the break in the latter stages.
The win hands Mauricio Pellegrino his first away win since he took charge on the south coast, while Hodgson faces an uphill battle to turn Palace's form around, especially with their next three matches pitting them against Manchester City, Manchester United and Chelsea.