Buckingham Palace confirmed earlier this month that traces of morphine had been found in an 'A' sample taken on the five-year-old after the Ascot Gold Cup - a race the mare famously won last year.
According to the Palace the positive sample was as a result of a contaminated feed product.
Estimate started as 2-1 favourite for the Goodwood Cup on Thursday on her first outing since the positive sample was announced, but faded badly in the final straight.
The Saeed bin Suroor-trained Cavalryman took the honours by powering home with two furlongs remaining.
Ridden by Kieren Fallon, the eight-year-old held off a late fightback from Ahzeemah, while last year's winner Brown Panther - part-owned by former footballer Michael Owen - was third.