Yes, in general the claim is as follows: given a weak equivalence of spaces $X \to X'$ under $A$, and given a fibration $B \to A$, then $X \times_A B \to X' \times_A B$ is a weak equivalence. This is a consequence of the right properness of the usual model structure on topological spaces (well, actually, it is essentially the same thing; the right properness can be seen because all objects are fibrant).
This is assuming you define homotopy pull-backs to mean that you replace one of the maps by a weakly equivalent fibration and form the pull-back with respect to that; in a general model category, you would have to replace all the maps by fibrations to get something invariant. (In fact, the whole point of homotopy limits and homotopy colimits is that ordinary limits and colimits are not homotopy invariant.)
A useful reference for this sort of thing is Dan Dugger's article on homotopy colimits.
The second half of the book by Bousfield and Kan, "Homotopy colimits, completions, and localizations" also treats this subject.