The win takes Southampton up to fifth with the majority of teams around them yet to play.
A first-team coach will be appointed after the departure of Sammy Lee, who joined the club with then manager Sam Allardyce last December. It is important to note that Palace was very unlucky to lose their last game as they were by far the better team.
Fraser Forster made a couple of important saves to ensure Southampton kept a clean sheet and this was the most pleasing aspect for manager Mauricio Pellegrino.
Moments later, the same player saw a strike deflected behind after a fine run, but Southampton soon sprang to the other end where Tadic wasted a crossing opportunity with a tame ball 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.
'Nervous, frantic' Crystal Palace leave Hodgson calling for confidence
The former Lazio man handled the threat of Benteke well and always seemed to be in the right position, while there were a few glimpses of his distribution skills with a handful of terrific left-footed passes from the back.
In truth the Saints could've easily scored two or three, with a dominant midfield performance from the side's trio of Oriol Romeu, Steven Davis and Mario Lemina.
Hodgson was keen to switch his focus to Crystal Palace and wants to put his England career to the back of his as he prepares for their crucial game against Southampton tomorrow.
Ipswich are the only previous side who failed to score in their opening five matches in a top-flight season (in 1970-71) - although they drew two of those five games.
Saints, who had last scored when they beat West Ham United 3-2 at home on August 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.
"I'm looking at Roy [Hodgson] as I'm watching the game and he doesn't inspire me".