In this paper, we propose a series of novel two-dimensional Janus structures TiSi$Z_3$H ($Z=$ N, P, As) and investigate their electronic, mechanical, piezoelectric, and transport properties using the density functional theory. The examined results demonstrated that Janus TiSi$Z_3$H monolayers are structurally stable. At the ground state, TiSiN$_3$H, TiSiP$_3$H, and TiSiAs$_3$H monolayers are all semiconductors with indirect bandgaps of 2.98, 1.06, 1.02~eV at the hybrid-functional level, respectively. The absence of vertical mirror symmetry in TiSi$Z_3$H results in the occurrence of both in-plane and out-of-plane piezoelectricity. Interestingly, TiSiAs$_3$H monolayer exhibits large out-of-plane piezoelectric response with coefficient $d_{31}$ up to $-0.45$~pm/V, that is appropriate for use in piezoelectric devices. The calculated results demonstrate that the carrier mobility in Janus TiSi$Z_3$H monolayers is high and directional anisotropic. Particularly, electron mobility in TiSiP$_3$H monolayer is observed up to 772.31~cm$^2$V$^{-1}$s$^{-1}$. Our results not only suggest and confirm some fundamental physical characteristics of new materials but also lay important premises for experimental studies as well as their application prospects in the future.