The nontrivial magnetic texture in real space gives rise to the intriguing phenomenon of the topological Hall effect (THE), which is relatively less explored in topological semimetals. Here, we report a large THE in the antiferromagnetic (AFM) state in single crystals of EuAgAs, an AFM Dirac semimetal. EuAgAs hosts an 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 an 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 semimetallic state can be realized under time-reversal symmetry breaking via an applied magnetic field. Our study establishes that EuAgAs is a system for exploiting the interplay of band topology and the topology of the magnetic texture.