The invention provides a device for taking out a broken rod of a threaded connecting rod piece through drilling and a construction method of the device. The device comprises a drill rod, and a drill bit or a rod fetching connector is detachably connected to the end of the drill rod. The drill rod is formed by sequentially and detachably connecting a set of rod sections, and the each rod section is sleeved with an annular limiting piece. During prestressed construction, when the rod piece is broken, due to the fact that a construction hole for the threaded connecting rod piece is narrow, namely the depth of the construction hole is over 3 m, however, the diameter of the construction hole is smaller than 50 mm, the broken rod left in the construction hole can hardly be taken out. By adoption of the device for taking out the broken rod through drilling, the broken rod left in the structure construction hole can be taken out, replacement of the threaded connecting rod piece is achieved, and hidden dangers during construction are eliminated; and meanwhile, maintenance cost is greatly reduced, and economic value is increased.