  We present the first results from a proof-of-concept search for dark sectors
via invisible decays of pseudoscalar $\eta$ and $\eta'$ mesons in the NA64h
experiment at the CERN SPS. Our novel technique uses the charge-exchange
reaction of 50 GeV $\pi^-$ on nuclei of an active target as the source of
neutral mesons. The $\eta, \eta' \to invisible$ events would exhibit themselves
via a striking signature - the complete disappearance of the incoming beam
energy in the detector. No evidence for such events has been found with
$2.9\times10^{9}$ pions on target accumulated during one day of data taking.
This allows us to set a stringent limit on the branching ratio ${\rm Br}(\eta'
\to invisible) < 2.1 \times 10^{-4}$ improving the current bound by a factor of
$\simeq3$. We also set a limit on ${\rm Br}(\eta \to invisible) < 1.1 \times
10^{-4}$ comparable with the existing one. These results demonstrate the great
potential of our approach and provide clear guidance on how to enhance and
extend the sensitivity for dark sector physics from future searches for
invisible neutral meson decays.
