Since the discovery of the Higgs boson in 2012, detailed studies of its properties have been ongoing. Besides its mass, its width—related to its lifetime—is an important parameter. One way to determine this quantity is to measure its off-shell production, where the Higgs boson mass is far away from its nominal value, and relating it to its on-shell production, where the mass is close to the nominal value. Here we report evidence for such off-shell contributions to the production cross-section of two Z bosons with data from the CMS experiment at the CERN Large Hadron Collider. We constrain the total rate of the off-shell Higgs boson contribution beyond the Z boson pair production threshold, relative to its standard model expectation, to the interval [0.0061, 2.0] at the 95% confidence level. The scenario with no off-shell contribution is excluded at a p-value of 0.0003 (3.6 standard deviations). We measure the width of the Higgs boson as ΓH=3.2+2.4−1.7MeV, in agreement with the standard model expectation of 4.1 MeV. In addition, we set constraints on anomalous Higgs boson couplings to W and Z boson pairs.