  A resonant state at $3.21^{+0.12}_{-0.04}$\,MeV, located just above the
one-neutron separation threshold, was observed for the first time in $^{12}$Be
from the $^{11}$Be\,$(d,p)^{12}$Be one-neutron transfer reaction in inverse
kinematics. This state is assigned a spin-parity of $0^-$, according to the
distorted-wave Born approximation (DWBA) and decay-width analysis. Gamow
coupled-channel (GCC) and Gamow shell-model (GSM) calculations show the
importance of the continuum-coupling, which dramatically influences the
excitation energy and ordering of low-lying states. Various exotic structures
associated with cross-shell intruding configurations in $^{12}$Be and in its
isotonic nucleus $^{11}$Li are comparably discussed.
