Koeman's side had not won, or scored, in their previous three Premier League outings, and it looked as though they were heading for a stalemate at St Mary's on Tuesday.
That was until Sadio Mane took advantage of Julian Speroni's spill to coolly clip home a finish with seven minutes remaining and send Southampton up to fifth.
"We didn't have that total confidence against Palace, who were very organised," Koeman told BBC Sport.
"It was tough to create and score. It wasn't the best game we have played this season, we were more lucky than in the last few weeks but they [Southampton's players] deserved this result.
"It's not easy when you are struggling as a football player, I know that. Finally, it is three points that makes the situation a little more comfortable.
"It was a crucial game to win today, now we have 10 days until the Chelsea game and it makes the situation a lot more comfortable and I hope they will get total confidence back and play like we did before."
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