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 decreases from x = 0.5 to 1. X-ray absorption spectroscopy has revealed that only Co is present in mixed valent 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 novel magnetic microstructure with unidirectional anisotropy that causes spontaneous exchange bias effect that can be tuned with the amount of antisite disorder.