They followed the 22m of line to find more than 816kg - worth approximately US$50million ($72.2m) - of cocaine in the debris field, the Sun-Sentinel reported.
Thetis was working in the Eastern Pacific as part of Operation Martillo when they found the turtle.
Operation Martillo (hammer) is a joint effort between the Coast Guard and international law enforcement partners to squash the drug trade.
While in the Eastern Pacific, Thetis helped seize 6755kg of cocaine and 6kg of marijuana in eight interdictions that led to the arrests of 24 suspected smugglers.
In total, the seized drugs had a street value of more than US$135m, the Coast Guard said.