We suggest that the exclusive Higgs + light (or b) jet production at the LHC,
$pp \to h+j(j_b)$, is a rather sensitive probe
of the light-quarks Yukawa couplings and of new physics (NP) in the
Higgs-gluon $hgg$ and quark-gluon $qqg$ interactions.
We study the Higgs $p_T$ distribution in $pp \to h+j(j_b)$, employing non-differential observables
to probe the different types of NP relevant for this process, which we parameterize
either as scaled SM couplings (the kappa-framework) and/or
through new higher dimensional effective operators (the SMEFT framework).
We find that the exclusive $h+j(j_b)$ production
at the 13 TeV LHC is sensitive to various NP scenarios,
with typical scales ranging from a few to O(10) TeV, depending
on the flavor, chirality and Lorentz structure of the underlying physics.