Quasi-coherent sheaves are closed under extensions

In $0\to \mathcal{F}\to \mathcal{G}\to \mathcal{E}\to 0$, assume the outer sheaves are quasi-coherent. On an affine open, any section of $\mathcal{E}$ has local lifts to $\mathcal{G}$ on a finite principal cover. Their differences are a Čech cocycle in the module defining $\mathcal{F}$. Exactness of the unit-ideal localization Čech complex makes this cocycle a coboundary, so adjusted lifts glue. Thus sections are exact on the affine open and on every principal open in it. Comparing the resulting sequences after localization, the outer isomorphisms force the middle localization map to be an isomorphism. Hence $\mathcal{G}$ is associated with its global-section module and is quasi-coherent.