Having confounded the naysayers after a summer of upheaval, things began to tail off recently for Saints as they struggled to turn possession into goals.
They toiled once again against Palace as a run of one win in five Barclays Premier League matches looked set to be extended, only for Sadio Mane to find the net and secure a late 1-0 win.
It ended a 387-minute wait for a goal at St Mary's and gave Southampton's top-four ambitions a timely shot in the arm, moving them to within a point of the Champions League places.
"Finally we score and we win at home," manager Koeman said, smiling.
"I can't say that we deserved the win today
Maybe after the first half, yes, but after the second half Crystal Palace did well.
"They had some good chances, had a shot off the post, but football is strange
Sometimes you play much better than we did today.
"I think it is all about the confidence, it is all about the confidence.
"We have the most difficult way to play football because we like to play football, we like to play offensive football.
"All the opponents are defending, like they did in the first half and in the second half they produced a bit more action to score.
"Then you need the total confidence to break it open and we don't have the confidence today
We struggled a little more than we like to but then it is all about a mental question for the players.
"But they keep the belief in the team, spirit in the team to try and score the goal and win the game.
"With the support we had from our fans, finally we have the three points and that makes the situation more comfortable."
It was Southampton's first home win in any competition since New Year's Day and a result on par with, if not bigger, than that victory against Arsenal.
"I think it can be very important," Koeman said
"I think this win maybe is one of the key moments of the season.
"We know we would like to fight for a high position in the table, but if you don't win three games at home in a row, you don't score (it makes it hard),
"It is all about confidence, it is all about a mental question for the players
It is not easy to change that
If you win, it all helps."
Southampton now have a much-needed break before travelling to Chelsea on March 15, while Palace host QPR the day before.
It is a match Alan Pardew's men will approach with a feeling of injustice as the Eagles boss claims Martin Atkinson, refereeing his first match since the controversial 1-1 draw between Chelsea and Burnley, should have awarded them a penalty for a foul on Yannick Bolasie.
"Obviously, the last game he was under a lot of pressure, but I think that was a penalty tonight," Pardew said.
"I just feel it was a penalty and when the centre-half has been as aggressive as Jose Fonte was tonight, he should give it.
"But, you know, we've been harshly treated I think by the officials in the last few games and sometimes you get a run like that."
Pardew's comments surprised those in the press conference and angered Southampton counterpart Koeman.
"No, it is so easy if you lose a game to tell that kind of story about referees," the Dutchman said.
"I think the referee was not involved in the final result - we scored and they didn't score and I know because I know that feeling.
"I think it is too easy and I think too many people are always talking about referees
I don't like that."
Source : PA