A search for pair-produced vector-like quarks using events with exactly one lepton ($e$ or $\mu$), at least four jets including at least one $b$-tagged jet, and large missing transverse momentum is presented. Data from proton-proton collisions at a centre-of-mass energy of $\sqrt{s} = 13$ TeV, recorded by the ATLAS detector at the LHC from 2015 to 2018 and corresponding to an integrated luminosity of 139 fb$^{-1}$, are analysed. Vector-like partners $T$ and $B$ of the top and bottom quarks are considered, as is a vector-like $X$ with charge +5/3, assuming their decay into a $W$, $Z$, or Higgs boson and a third-generation quark. No significant deviations from the Standard Model expectation are observed. Upper limits on the production cross-section of $T$ and $B$ quark pairs as a function of their mass are derived for various decay branching ratio scenarios. The strongest lower limits on the masses are 1.59 TeV assuming mass-degenerate VLQs and branching ratios corresponding to the weak-isospin doublet model, and 1.47 TeV (1.46 TeV) for exclusive $T \rightarrow Zt$ ($B/X \rightarrow Wt$) decays. In addition, lower limits on the $T$ and $B$ quark masses are derived for all possible branching ratios.