The first evidence for off-shell Higgs boson production is reported in the final state with two Z bosons decaying into either four charged leptons (muons or electrons), or two charged leptons and two neutrinos, and a measurement of the Higgs boson width is performed. Results are based on data from the CMS experiment at the LHC at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of up to 140 fb$^{-1}$. The total rate of off-shell Higgs boson production beyond the Z boson pair production threshold, relative to its standard model expectation, is constrained to the interval [0.0061, 2.0] at 95% confidence level. The scenario with no off-shell production is excluded at 99.97% confidence level (3.6 standard deviations). The width of the Higgs boson is extracted as $\Gamma_{\mathrm{H}{}} =$ 3.2$_{-1.7}^{+2.4}$ MeV, in agreement with the standard model expectation of 4.1 MeV. The data are also used to set new constraints on anomalous Higgs boson couplings to W and Z boson pairs.

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