Exchange bias effect in the ferromagnetic double perovskite compound Y\(_2\)CoMnO\(_6\), which is also a multiferroic, is reported. The exchange bias, observed below 8~K, is explained as arising due to the interface effect between the ferromagnetic and antiferromagnetic clusters created by {\it antisite} disorder in this material. Below 8~K, prominent ferromagnetic hysteresis with metamagnetic "steps" and significant coercive field, \(H_c \approx\) 10~kOe are observed in this compound which has a \(T_c \approx\) 75~K. A model based on growth of ferromagnetic domains overcoming the elastic energy of structurally pinned magnetic interfaces, which closely resembles martensitic-like transitions, is adapted to explain the observed effects. The role of {\it antisite} disorder in creating the domain structure leading to exchange bias effect is highlighted in the present work.