The 29-year-old, from Hastings, finished the race in a time of 48.28 seconds.
Ukraine's Vladyslav Yepifanov took the gold and Great Britain's Jack Eyers took silver.
The podium place sees the New Zealand team finish the Games with nine medals: One gold, four silver and four bronze medals.
Cowan said it was "something big and special" to win a Paralympics medal.
"The preparation building up to this was the hardest I've had on and off the water," he said.
"It was challenging along the way, since World Champs I've had a number of family events happen. I've lost three whānau members, and I had to take time off for that.
"But when it came to training, I was able to turn up and do my thing. Six weeks in Italy was really good to lock in and my coach [Myka Nuku] and I worked off our race plan from world champs, especially in these conditions. It gave us good intel on how to prepare for this heavy side wind we had in the final.
"To [win bronze] in a giant field of quality paddlers was just unreal and I take my hat off to all the guys in our division. They are all strong paddlers, but I had full belief that the right result would come if I just focused on my own performance."
Cowan was badly hurt at the age of 15 when he was clipped by a car while training for the IronMāori triathlon, and later had his injured leg amputated.
He later discovered paddle sports and has since competed internationally in both waka ama and para canoe races.
The Paris Games were his first Paralympics.
New Zealand Paralympic Team Chef de Mission Raylene Bates said the athletes had been "amazing" in Paris.
"They've been resilient and given their all in the competition, we couldn't ask for more."
The 24-strong team set one Paralympic record, four Oceania records, 11 national records and 12 personal best performances between them, she said.
"It was refreshing to see so many personal bests from our athletes and a number of memorable performances.
"Yes, there have been disappointments, which is sport, but I believe that is more a reflection of rising performance standards across the board and the sheer number of world records and Paralympic records which have been set at Paris 2024.
The New Zealand athletes had six fourth places between them "which could have been medals", she said.
"Anna Grimaldi was 1cm off the podium in the long jump and Will Stedman was 2cm off the podium in the men's long jump. Cam Leslie narrowly missed out in a medal in the pool with two fourth places. What we have seen at these Games is a changing of the guard in which young athletes have emerged to snatch medals from our established names.
"Our challenge is to meet these rising standards in New Zealand and continue to build our programmes to set us up for continued success at Los Angeles 2028 and Brisbane 2032."
