Let $\mathscr{L}$ be a holomorphic line bundle (perhaps smooth suffices, but I'm concerned with the holomorphic case anyway) on a smooth complex manifold $X$. We can define its Chern class via the exponential exact sequence: $$ 0 \rightarrow \mathbb Z \rightarrow \mathscr O_X \rightarrow \mathscr O_X^\times \rightarrow 0 $$ And the associated cohomology sequence which includes: $$ H^1(X, \mathscr O_X^\times) = \text{Pic}(X) \rightarrow H^2(X, \mathbb{Z}) $$ We can use this map as a definition for the Chern class $c_1(\mathscr L)$.
It is true that if $\sigma$ is a holomorphic section of $\mathscr L$, then if we let $[\sigma]$ denote the homology class represented by the vanishing locus of $\sigma$ (the points $p$ where $\sigma(p) \in \mathscr L$ is $0$), then $c_1(\mathscr L)$ is Poincaré dual to $[\sigma]$ (at least in the case that the vanishing locus of $\sigma$ is a irreducible analytic subvariety).
Does anyone know (or know a reference for) a straightforward proof of this fact? Griffiths-Harris shows it via the curvature form definition of Chern class, but this is more hard differential geometry than I am comfortable with. I'd love a proof directly from this exact sequence, but I'd also be happy with assuming that $c_1(\mathscr L)$ as defined here agrees with the topologically-defined Chern class (i.e. defined by axioms or the Euler class or by pullback from a Grassmanian).