Cox, 43, has been out of work since leaving Mansfield in November, and he was in charge at Field Mill when the Stags made their return to the Football League at the end of the 2012-13 season.
The Gulls struggled badly after being relegated to the Vanarama Conference Premier, with Hargreaves unable to hold on to the momentum from a fast start to the season.
They slumped down the table and, at one point, were deep in relegation trouble before a run of late-season victories lifted them into 13th.
Torquay were taken over by a new consortium late last week and placing Hargreaves on gardening leave, along with assistant Lee Hodges and goalkeeping coach Kenny Veysey, was one of their first moves.
Cox has a tough rebuilding job on his hands, with Torquay only having nine players on their books three weeks before they kick off their pre-season campaign – four of those players are teenagers.