Cooper had to settle for second place, his best world cup finish at elite level, and was delighted with his performance.
"I think that was the best race I've ever ridden with the tightest finish too," Cooper said on twitter. "It was a real thriller and great to be racing at the very front of an elite world cup."
Gaze, who won the short track XCC race on Saturday, took the lead from the front row of the grid on the opening lap, but a front tyre puncture and crash on the rock section forced the Commonwealth Games champion out of the race.
It is Cooper's best finish in a World Cup, after the Trek Factory rider finished third in Germany last year, while he was ninth at Nove Mesto in the opening round in 2017.
Shurter leads the world cup standings from Marotte with Gaze fourth and Cooper fifth.
Subway National Hub rider Ben Oliver was 10th in the Under-23 elite race after a strong ride, pushing up with a chase bunch to close in on the leaders mid-race before Romania's Vlad Dascalu ramped up the pace on the final lap.
Oliver is sixth on the overall standings after three rounds, with five weeks until the next round in Italy.
Oceania champion Samara Sheppard was 50th in the elite women's race won by Denmark's Annika Langvad in a sprint finish with Switzerland's Jolanda Neff, while fellow Kiwi Kate Fluker did not finish.
Results:Elite men: Nino Schurter (SUI) 1:21:31, 1; Anton Cooper (NZL, Trek Factory Team) same time, 2; Maxime Marotte (FRA) at 8sec, 3. Also: Sam Gaze dnf.
Elite women: Annika Langvad (DEN) 1:20:37, 1; Jolanda Neff (SUI) at 1s, 2; Pauline Prevot (FRA) at 47s, 3. Also: Samara Sheppard (NZL) at 11:43, 50; Kate Fluker (NZL) dnf.
Under-23 men: Vlad Dascalu (ROU) 1:23:44, 1; Filippo (SUI) at 21s, 2; Alan Hatherly (RSA) at 24s, 3. Also: Ben Oliver (NZL) at 1:19, 10; Paul Wright (NZL) dnf.