Eaton's body was found around 60 meters inside the cave beneath an air shaft which had been covered by a large wooden pallet, Lagoudakis added.
Eaton was attending a conference at the Orthodox Academy in Crete and is believed to have disappeared during a run.
The police said on Wednesday that Eaton was asphyxiated. Minor staff were also found on her body, but police said they were not believed to be the cause of her death. The police believe the body was dumped inside the cave, because it was found face down.
Lagoudakis told CNN on Thursday that he had seen a case like this in his four years as police chief.
Her family initially believed Eaton, a regular runner, probably died during a run of heat exhaustion or a fall in the rough terrain.
The horrific details surrounding Eaton's death have shocked the locals, some of whom thought she died in a hiking accident. Crete is known to be among the safest of the Greek islands despite its size and the large number of tourists.
"We are deeply shocked and disturbed by this tragic event," the Planck Institute at Dresden University in Germany, Eaton's employer, said in a statement. "Suzanne was an outstanding and inspiring scientist, a loving spouse and mother, and athlete as well as a truly wonderful person beloved to us all." Eaton was the wife of British scientist Tony Hyman and mother of two sons, according to Institute.