By analyzing $6.32~mathrm{fb}^{-1}$ of $e^+e^-$ annihilation data collected at the center-of-mass energies between 4.178 and 4.226,GeV with the BESIII detector, we determine the branching fraction of the leptonic decay $D_s^+totau^+ u_tau$ with $tau^+topi^+pi^0bar u_tau$, to be $mathcal{B}_{D_s^+totau^+ u_tau}=(5.29pm0.25_{rm stat}pm0.20_{rm syst})%$. We estimate the product of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{cs}|$ and the $D_s^+$ decay constant $f_{D^+_s}$ to be $f_{D_s^+}|V_{cs}|=(244.8pm5.8_{rm stat}pm4.8_{rm syst})~mathrm{MeV}$ using the known values of the $tau^+$ and $D_s^+$ masses as well as the $D_s^+$ lifetime, together with our branching fraction measurement. Combining with the value of $|V_{cs}|$ obtained from a global fit in the standard model and $f_{D_s^+}$ from lattice quantum chromodynamics, we obtain $f_{D_s^+}=(251.6pm5.9_{rm stat}pm4.9_{rm syst})$,MeV and $|V_{cs}| = 0.980pm0.023_{rm stat}pm0.019_{rm syst}$.
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