Air traffic at Sabiha Gokcen was halted throughout the incident but had returned to normal after the man's arrest.
Mehmet Tutan, one of the passengers, told reporters at Sabiha Gokcen that people on board also believed they had landed in Sochi but realized the plane was in Turkey after switching on their mobile phones.
"We thought we had landed in Sochi. We thought so for a long time," Tutan said. "Then we saw that there was a (Turkish) network on our mobile phone, that we were able to call outside, and that there was no roaming, we understood that we weren't in Sochi."
The plane landed at about 6 p.m. Turkish time, just as the opening ceremony for the Olympics was about to begin. The executive creative director of the Olympics opening ceremony told reporters afterward he heard of the threat but didn't alter the show's plans in any way.
"We had so much adrenaline in our veins that we could not grasp much," Konstantin Ernst said through an interpreter.
With about 100,000 police, security agents and army troops flooding Sochi, Russia has pledged to ensure "the safest Olympics in history." But terror fears fueled by recent suicide bombings have left athletes, spectators and officials worldwide jittery about potential threats.
"It would be wrong to make any comment before all the facts are known but any security questions are of course a matter for the authorities," International Olympic Committee spokesman Mark Adams said.
Security experts warn that Islamic militants in the Caucasus, who have threatened to derail the Winter Games that run from Feb. 7-23, could achieve their goal by choosing soft targets away from the Olympic sites or even outside Sochi.
Olympic organizers introduced blanket screening of all visitors, requiring them to share passport details to get a Winter Games spectator pass. Officials also cut access to vehicles lacking Sochi registration or a special pass, and guards were searching all train commuters.
Associated Press writer Steve Wilson in Sochi contributed to this report.