The 2011 world champion had enough strength to hold off Nino Schurter of Switzerland in 1:29:07, with Italy's Marco Fontana third, 25 seconds adrift.
Julien Absalon of France, the 2004 and 2008 champion, did not finish after falling down the field following a puncture.
Great Britain's Liam Killeen, who was fifth in Athens 2004 and seventh in Beijing four years later, crashed out on the second of seven laps, tumbling at Deane's Drop.
Absalon lost touch with Schurter, who won four rounds and the overall World Cup series title this season, and Kulhavy on the first lap.
Schurter, Kulhavy and Fontana established a narrow lead at the end of the opening circuit, with Jose Antonio Hermida of Spain leading the chase.
Killeen tumbled down Deane's Drop, a steep boulder section, and was joined in failing to finish lap two by Absalon.
South Africa's Burry Stander and Hermida bridged the gap at the end of the third lap to join the three leaders.
Kulhavy took the lead on the fifth lap and stretched the group with the relentless pace, opening up a narrow lead over Schurter and Fontana with two laps to go, with Stander and Hermida losing contact.
Kulhavy and Schurter were vying for the win, with Fontana dropped in the closing stages of the final lap. Schurter moved to the front but Kulhavy rounded his rival to triumph.