A direct search for Higgs bosons produced via vector-boson fusion and subsequently decaying into invisible particles is reported. The analysis uses 139 $fb-1$ of $pp$ collision data at a centre-of-mass energy of $s$=13 $TeV$ recorded by the ATLAS detector at the LHC. The observed numbers of events are found to be in agreement with the background expectation from Standard Model processes. For a scalar Higgs boson with a mass of 125 $GeV$ and a Standard Model production cross section, an observed upper limit of $0.145$ is placed on the branching fraction of its decay into invisible particles at 95% confidence level, with an expected limit of $0.103$. These results are interpreted in the context of models where the Higgs boson acts as a portal to dark matter, and limits are set on the scattering cross section of weakly interacting massive particles and nucleons. Invisible decays of additional scalar bosons with masses from 50 $GeV$ to 2 $TeV$ are also studied, and the derived upper limits on the cross section times branching fraction decrease with increasing mass from 1.0 $pb$ for a scalar boson mass of 50 $GeV$ to 0.1 $pb$ at a mass of 2 $TeV$.