This is a theorem of Lang's from 1956. Here's an online document giving a proof (in the form $H^1(A,k)=0$):

Lecture 14: Galois Cohomology of Abelian Varieties over Finite Fields, 
William Stein. http://wstein.org/edu/2010/582e/lectures/582e-2010-02-12/582e-2010-02-12.pdf

Stein notes that there is a "more modern proof" in the first few sections of 
Chapter  VI  of  Serre's
*Algebraic  Groups  and  Class  Fields*.

The original article is Serge Lang, "Abelian varieties over finite fields," *Proceedings of the National Academy of Sciences* **41.3** (1955): 174-176. It is available at http://www.pnas.org/content/41/3/174.short. It's very short, but phrased in the "old-style" Weil language of algebraic geometry.