In his first season in charge, he led the Saints to a seventh-place finish with 60 points, their highest points tally in the Premier League until it was broken in the final game of the current campaign.
"It's a fantastic achievement for everybody in the club," Koeman told British media.
"It's been an even better season than last year. A record of points is crazy and full compliments to everybody at Southampton because the support of the fans was amazing and the spirit and belief of the players is very enjoyable for the manager."
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The 53-year-old, who has a year left on his contract, said he has no desire to leave the club.
"I still have a one-year contract. I spoke to the board and we had the meeting and it's always my wish to stay because I like the club, fans and players," Koeman added.
"We understand every time it's more difficult because we need to improve, and expectation will be higher than it was two years ago. If we have the same ambition again, then no problem."
Southampton, tied on 63 points with Manchester United, will have to wait for United to play Bournemouth on Tuesday to learn if they will keep their hold on fifth place, which gives them direct qualification to the Europa League group phase.
United's match against Bournemouth was also scheduled on Sunday, but the game was abandoned after security personnel found a suspect package at Old Trafford.
(Reporting by Ian Rodricks in Bengaluru; editing by Sudipto Ganguly)
