The non-trivial magnetic texture in real space gives rise to the intriguing phenomenon of topological Hall effect (THE), which is relatively less explored in topological semimetals. Here, we report large THE in the antiferromagnetic (AFM) state in single crystals of EuAgAs, an AFM Dirac semimetal. EuAgAs hosts AFM ground state below $T_N$ = 12 K with a weak ferromagnetic component. The in-plane isothermal magnetization below $T_N$ exhibits a weak metamagnetic transition. We also observe chiral anomaly induced positive longitudinal magnetoconductivity which indicates a Weyl fermion state under applied magnetic field. The first-principles calculations reveal that EuAgAs is an AFM Dirac semimetal with a pair of Dirac cones, and therefore, a Weyl semimetailic state can be realized under time-reversal symmetry breaking via an applied magnetic field. Our study establishes that EuAgAs is a novel system for exploiting the interplay of band topology and the topology of the magnetic texture.