You can find a similar relationship in equation (3) of the following paper

http://users.eecs.northwestern.edu/~dguo/publications/Guo09ISIT.pdf

The authors refer to

S. Kullback, Information Theory and Statistics. New York: Dover, 1968.

for a proof of this resutl.

You can find a similar relationship (for a one-dimensional parameter) in equation (3) of the following paper

D. Guo (2009), Relative Entropy and Score Function: New Information–Estimation Relationships through Arbitrary Additive Perturbation, in Proc. IEEE International Symposium on Information Theory, 814–818. (stable link).

The authors refer to

S. Kullback, Information Theory and Statistics. New York: Dover, 1968.

for a proof of this result.