The exchange bias effect at the magnetic in terfaces and multi-magnetic phases strongly depends on the antisite disorder (ASD) driven spin configuration in the double perovskite systems. The percentage of ASD in double perovskites is extensively accepted as a key for designing diverse new nanospintronics with tailored functionalities. In this regards, we have investigated such ASD driven phenomena in Ca²⁺ doped bulk and polycrystalline La_2−xCa_xCoMnO₆ (0 ≤ x ≤ 1) series of samples. The structural and Raman studies provide evidence of an increase in the disorder due to the increment of Ca concentration in the parent compound (x = 0). The enhancement of disorder in the doped system induces various magnetic orderings, magnetic frustration and cluster glass-like behavior, which have been confirmed from AC and DC magnetic studies and neutron diffraction studies. As a result, significantly large exchange bias effects, namely zero-field cooled (spontaneous) and field-cooled (conventional) exchange bias, are found. These results reveal the tuning of ASD by doping, which plays an active role in the spin configuration at the magnetic interfaces.