In this paper, we report on an experimental investigation of a shape-controlled percolation transition in two-dimensional (2D) amorphous packing of dimers without long-range order. In the maximally random jammed (MRJ) packing of asymmetric dimers consisting of head and body, a dramatic increase in the connectivity of heads upon increasing the head-to-body size ratio. leads to a percolation transition of the heads at the well-defined percolation threshold. In comparison with binary disks, the existence of a bond in dimers causes the heads to be homogeneously distributed over a system by inhibiting the local segregation. Interestingly, we found, however, that the cluster structure at the percolation threshold is insensitive to the bond, even though the existence of the bonds affects the percolation threshold as well as the head distribution. The fractal dimensions at the percolation threshold obey the universal law of the 2D percolation theory independently of the existence of bonds. Our finding can provide us with a new perspective of interesting applications of randomly assembled binary composites by using the homogeneous particle distribution and the sensitively tunable connectivity under particle shape control.