The 29-year-old won through her women's under-52 kg pool before losing to France's Amandine Buchard in the semi-finals by ippon on Monday (11th January).
Buchard went on to win her final against Shishime Ai of Japan through an ippon, while Kelmendi beat Charline van Snick in the repechage to clinch bronze.
After winning his three pool matches, An faced two bouts that both went to golden score.
In the semi-finals, An and Moldovan Denis Vieru went to five minutes 39 seconds of extra sudden-death competition before Vieru received his third shido warning for non-combativity and the Korean was awarded the victory.
Israel's Baruch Shmailov took An to extra time in the final too, before An pulled off a takedown 2:23 into golden score and was awarded a waza-ari score, ending the bout.
Two-time women's under-48 kg world champion Daria Bilodid was beaten in the semi-finals as she clinched bronze in her weight class.
Japanese star Tonaki Funa, who lost to Bilodid in the world championship finals of 2018 and 2019, beat the Ukrainian by ippon in Qatar, before going on to lose to eventual winner Distria Krasniqi of Kosovo.
Bilodid beat Catarina Costa of Portugal by ippon in the repechage.
Women's under-57 kg world number one Jessica Klimkait of Canada also leaves Qatar with bronze after being defeated by Sarah Cysique of France by waza-ari in the semi-finals.
While Yoshida Tsukasa of Japan would go on to beat Cysique in the final, Klimkait – who has to beat defending world champion Christa Deguchi to take Canada's only Tokyo 2020 spot in the weight class – hauled Timna Nelson Levy of Israel down for an ippon win in the repechage.
The day's other event, the men's under-60 kg, was won by Korean Kim Won-jin.