  We explore the emergence of gravitation from entanglement in holographic CFTs
with gravitational anomalies. More specifically, the holographic correspondence
between topologically massive gravity (TMG) with gravitational Chern-Simons
term in the 3D bulk and its dual CFT with unbalanced left and right moving
central charges on the 2D boundary, is studied from the quantum entanglement
perspective. Using the first law of entanglement, we derive the holographic
dictionary of the energy-momentum tensor in TMG, including the chiral case with
logarithmic mode. Furthermore, we show that the linearized equation of motion
of TMG can also be obtained from entanglement using the Wald-Tachikawa
covariant phase space formalism. Finally, we identify a quasi-local
gravitational energy in the entanglement wedge as the holographic dual of
relative entropy in gravitationally anomalous CFTs. The positivity and
monotonicity of relative entropy imply that such a gravitational energy should
be positive definite and become larger when increasing the size of the
entanglement wedge. These constraints from quantum information may be
potentially used to discuss the UV inconsistent issues of TMG.
