It came on a dramatic afternoon in the velodrome after Cox – who has multiple sclerosis and suffered a calf injury in the build-up to the Games – had to be helped off the track by support staff.
The 33-year-old was not allowed to restart, as new rules brought in before these Games only permit a second attempt if there is a mechnical fault with a bike.
Dutch rider Caroline Groot won the first gold of the Games in a time of 35.566 seconds.
Hosts France picked up their first medal as Marie Patouillet took silver, with Canada’s Katie O’Brien winning bronze.
Reigning world champion Cox had looked strong in qualifying, recording 35.436 seconds – the second quickest overall after Groot, and ultimately faster than the time the Dutch athlete rode in the medal final.
Cox has previously been dominant in this event, having won gold at the 2016 and 2020 Paralympics – but found only heartbreak here.
She was visibly emotional and unsteady on her feet as she was helped from the track, before being embraced and comforted by GB staff.
However, BBC Sport has been told that Cox has not sustained a significant injury, whether new or to her previously problematic calf.
This means she may be able to come back and compete for a medal later in the Games, with the mixed C1-5 team sprint taking place on Sunday.