Doping at the rare-earth site by divalent alkaline-earth ions in perovskite lattice has witnessed a variety of magnetic and electronic orders with spatially correlated charge, spin and orbital degrees of freedom. Here, we report an antisite disorder driven spontaneous exchange bias effect as a result of hole carrier (Sr2+) doping in La2-xSrxCoMnO6 (0 < x < 1) double perovskites. X-ray diffraction and Raman spectroscopy have evidenced an increase in disorder with the increase of Sr content up to x = 0.5 and thereby a decrease from x = 0.5 to 1. X-ray absorption spectroscopy has revealed that only Co is present in the mixed valence of Co2+ and Co3+ states with Sr doping to compensate the charge neutrality. Magnetotransport is strongly correlated with the increase of antisite disorder. The antisite disorder at the B-site interrupts the long-range ferromagnetic order by introducing various magnetic interactions and instigates reentrant glassy dynamics, phase separation and canted type antiferromagnetic behavior with the decrease of temperature. This leads to a novel magnetic microstructure with unidirectional anisotropy that causes a spontaneous exchange bias effect that can be tuned with the amount of antisite disorder.