The Lefschetz fixed point theorem states implies that the Lefschtz number of any map homotopic to the identity is the Euler characteristic of of the manifold. Suppose that $X$ is a smooth vector field with nondegenerate zeros. Denote by $\Phi_X^t$ the flow generated by $X$. Then for $t>0$ the fixed points of $\Phi_X^t$ are precisely the zeros of of $X$. The Lefschetz number of $\Phi_X^t$ is then the sum of the indices of the zeros of $X$. On the other hand, $\Phi_X^t$ is obviously homotopic to the identity and thus its Lefschetz number is the Euler characteristic which must be equal to the sum of indices of the zeros of $X$. For more details see Section 7.3. of my book, Lectures on the Geometry of manifolds, http://www3.nd.edu/~lnicolae/Lectures.pdf

Thanks Liviu, I'll back with some comments!