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We like to do more than that, actually. The B-field is an element in the differential cohomology class $\check{H}^3(M)$, or, more geometrically, a connection on an abelian gerbe. Thus, there is a class $[H] \in H^3(M,Z)$ characterizing the gerbe. In the B-model, this twists the derived category. The connection is the part that changes the A-model, and when $[H] = 0$, you exactly get that the differential cohomology group is $H^2(X,U(1))$. In the geometric language, it's a flat connection on a trivial gerbe.