In the present work we demonstrate that C-doped Zr$_{5}$Pt$_{3}$ is an electron-phonon superconductor (with critical temperature T$_mathrm{C}$ = 3.7,K) with a nonsymmorphic topological Dirac nodal-line semimetal state, which we report here for the first time. The superconducting properties of Zr$_{5}$Pt$_{3}$C$_{0.5}$ have been investigated by means of magnetization and muon spin rotation and relaxation ($mu$SR) measurements. We find that at low temperatures the depolarization rate is almost constant and can be well described by a single-band $s-$wave model with a superconducting gap of $2Delta(0)/k_mathrm{B}T_mathrm{C}$ = 3.84, close to the value of BCS theory. From transverse field $mu$SR analysis we estimate the London penetration depth $lambda_{L}$ = 469 nm, superconducting carrier density $n_{s}$ = 2$times$10$^{26}$ $m^{-3}$, and effective mass m$^{*}$ = 1.584 $m_{e}$. Zero field $mu$SR confirms the absence of any spontaneous magnetic moment in the superconducting ground state. To gain additional insights into the electronic ground state of C-doped Zr$_5$Pt$_3$, we have also performed first-principles calculations within the framework of density functional theory (DFT). The observed homogenous electronic character of the Fermi surface as well as the mutual decrease of $T_mathrm{C}$ and density of states at the Fermi level are consistent with the experimental findings. However, the band structure reveals the presence of robust, gapless fourfold-degenarate nodal lines protected by $6_{3}$ screw rotations and glide mirror planes. Therefore, Zr$_5$Pt$_3$ represents a novel, unprecedented condensed matter system to investigate the intricate interplay between superconductivity and topology.
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