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We study the temporal evolution of the mutual information (MI) in a one-dimensional Kitaev chain, coupled to a fermionic Markovian bath, subsequent to a global quench of the chemical potential. In the unitary case, the MI (or equivalently the bipartite entanglement entropy) saturates to a steady-state value (obeying a volume law) following a ballistic growth. On the contrary, we establish that in the dissipative case the MI is exponentially damped both during the initial ballistic growth as well as in the approach to the steady state. We observe that even in a dissipative system, postquench information propagates solely through entangled pairs of quasiparticles having a finite lifetime; this quasiparticle picture is further corroborated by the out-of-equilibrium analysis of two-point fermionic correlations. Remarkably, in spite of the finite lifetime of the quasiparticles, a finite steady-state value of the MI survives in asymptotic times which is an artifact of nonvanishing two-point correlations. Further, the finite lifetime of quasiparticles renders to a finite length scale in these steady-state correlations.