Christian Eriksen scored the only goal in a 1-0 Premier League win for the hosts five minutes before the break at White Hart Lane, but Mane should have done better when Ryan Bertrand played in a dangerous ball from the left late on.
The Senegal international could not sort his feet out in the six-yard box and the chance to pull his side level went begging.
But Koeman refused to be too hard on the man who finally completed his move from Red Bull Salzburg last month.
"That's part of football," he said. "Sometimes you score, sometimes not.
"He [Mane] got two good chances in this game. In the first half a good one as well, but maybe the best one was in the second half.
"He was playing good, he was fast, he was doing his best. Missing chances is part of football."
Sunday's defeat brought an end to Southampton's seven-match unbeaten run in all competitions, but Koeman was still pleased with his side's overall performance.
"I'm disappointed about the final result, but not disappointed about our performance," he added. "We knew it was difficult, because in the first half they did some great pressing.
"It was not easy to create possibilities or opportunities.
"It was a pity because they scored before half-time. We spoke at half-time about keeping good organisation in the second half and little by little we had to push up a bit more and I think we did that very well."
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