The Saints had failed to find the net in their own backyard for over six hours before Mane struck seven minutes from time.
A relieved Koeman said: “Finally we win at home and score a goal. I can’t say we deserved it. Crystal Palace had some good chances.
“But football is strange, we have played better than today. This win is maybe one of the key moments of the season.
“We know it will be very difficult to qualify for the Champions League, but we will try to do anything to finish in the highest position in the table.
“I hope we will have the confidence after this win and then everything is possible.”
Palace boss Alan Pardew said: “We should have had a penalty in the second half. Jose Fonte had been aggressive all night and was over-aggressive on Yannick Bolasie.
“The last game the referee (Martin Atkinson) was under a lot of pressure and it was a penalty tonight.”
But Koeman added: “It’s so easy if you lose a game to tell that kind of story about referees. I think the referee was not involved in the final result.
“We scored and they didn’t score. I think people are talking about referees and I don’t like that.”
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Crystal Palace are still in the relegation battle
Mane followed up after Palace keeper Julian Speroni failed to hold James Ward-Prowse’s shot.
For Palace, it came as a bitter blow after they had spurned the better opportunities.The Saints started like a team anxious to get back to their winning ways, but they lacked the confidence to turn their dominance into clear chances.
They almost undid all their good work when goalkeeper Fraser Forster spilled a 36th-minute cross and was forced to react quickly to prevent Wilfried Zaha scoring.
Southampton had barely troubled Speroni, while at the other end Dwight Gayle had seen a penalty appeal turned down. Atkinson’s thoughts must have flashed back to his recent penalty problems at Stamford Bridge, when he ignored Palace’s claims after Gayle went down under a challenge from Fonte.
The best chance of the first half fell to Graziano Pelle, who should have ended his woeful run for Saints when he stabbed wide from close range after Speroni parried Eljero Elia’s shot. Then the Saints had an escape when Zaha fired against a post.
Southampton should have gone ahead when Maya Yoshida stabbed wide of an open goal before the late breakthrough sank Palace.
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