It is true if the projective dimension of $M$ over $A$ is at most $3$, and counter examples exist when the projective dimension is $4$. 

The first counter example was given in Lucho Avramov's [paper](http://www.jstor.org/pss/2374187)
"Obstructions to the existence of a multiplicative structure on minimal resolutions". A simplified example, ($A=k[t_1,t_2,t_3,t_4], M= A/(t_1^2, t_1t_2, t_2t_3, t_3t_4, t_4^2)$, here you can choose the $x_i$s to be  any regular sequence in the annihilator of $M$)
 together with discussion of related and more recent  results can be found in Section 2 of this [note](http://books.google.com/books?id=C5j4xCl8Q80C&lpg=PA1&ots=F6RUMrczXS&dq=Infinite%20free%20resolutions&pg=PA1#v=onepage&q=Infinite%20free%20resolutions&f=false) (available on Avramov's [website](http://www.math.unl.edu/~lavramov2/papers.html)).