Let $E\to B$ be a fibration with fiber F, and assume for simplicity that B is connected. Suppose moreover that B and F have Euler characteristics (perhaps they are manifolds). Then often, one can conclude that E has an Euler characteristic as well, and that
$$ \chi(E) = \chi(B)\cdot \chi(F). $$
The only proof of this that I have been able to find uses a spectral sequence argument, and requires that $\pi_1(B)$ act trivially on the homology of F, so that the homology in the spectral sequence can be taken with constant coefficients. This condition is sometimes referred to as orientability of the fibration (with respect to the homology theory, normally rational homology).
Is the result known to be true any more generally than this? Is there any other known proof? Are there any examples where it is known to be false?