A precision measurement of the $Z$ boson production cross-section at $s=13$ TeV in the forward region is presented, using $pp$ collision data collected by the LHCb detector, corresponding to an integrated luminosity of 5.1 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$. The production cross-section is measured using $Z\to \mu +\mu -$ events within the fiducial region defined as pseudorapidity and transverse momentum $pT>20$ GeV/$c$ for both muons and dimuon invariant mass GeV/$c2$. The integrated cross-section is determined to be \begin{equation*} \sigma(Z\rightarrow\mu^+\mu^-) = 196.4 \pm 0.2 \pm 1.6 \pm 3.9~pb, \end{equation*} where the first uncertainty is statistical, the second is systematic, and the third is due to the luminosity determination. The measured results are in agreement with theoretical predictions within uncertainties.