Well. Even in the case of a DG (or $A_\infty$) algebra $A$, infinitesimal (i.e. 1st order) deformations are classified by $HH^2(A,A)$. Namely, the structure maps (a-k-a Taylor components) of an $A_\infty$-algebra, viewed as elements of the Hochschild cochain complex, do have total degree $2$.
DamienC
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