The 22-year-old, from Stainforth twice came from behind to lead 28-26 with seconds remaining in the final round.
But an impressive Rashitov fought back again to win a topsy-turvy contest 34-29.
Sinden, the 2019 world champion, was bidding to become Britain’s first ever male Olympic taekwondo champion.
"It was my gold medal to give away - obviously he is a good fighter, I just made a few mistakes,” he said.
"I think I got unlucky with a few things as well but that is taekwondo.
"I thought he was on the back foot, you have to commend him for what he did - a few mistakes from me, well done to him, you'll see me again in Paris.
"Maybe eventually I will get over it, but for now it has got me that I didn't win gold when I think it was there for me to take,” he added.
“I will take that and improve on it."
Sinden had coasted through his first two matches in impressive fashion earlier in the day before a hard fought win against China's Zhao Shuai in the semi-final.
He trailed 19-12 in the final round against Shuai, a gold medallist in the lighter -58kg category in Rio in 2016, but produced a stunning late onslaught to win 33-25.
Sinden eased to a 53-8 victory against New Zealand’s Tom Burns in the round of 16 before a 39-19 win over Turkey’s Recber Hakan in the quarter-finals.