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_-1.7 MeV, 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.