  We study the production of a Higgs boson in association with bottom quarks
($b\bar{b}H$) in hadronic collisions at the LHC, including the different
contributions stemming from terms proportional to the top-quark Yukawa coupling
($y_t^2$), to the bottom-quark one ($y_b^2$), and to their interference ($y_b
y_t$). Our results are accurate to next-to-leading order in QCD, employ the
four-flavour scheme and the (Born-improved) heavy-top quark approximation. We
find that next-to-leading order corrections to the $y_t^2$ component are
sizable, making it the dominant production mechanism for associated $b\bar{b}H$
production in the Standard Model and increasing its inclusive rate by almost a
factor of two. By studying final-state distributions of the various
contributions, we identify observables and selection cuts that can be used to
select the various components and to improve the experimental sensitivity of
$b\bar{b}H$ production on the bottom-quark Yukawa coupling.
