  We present a new method to evaluate the $\alpha'$-expansion of genus-one
integrals over open-string punctures and unravel the structure of the elliptic
multiple zeta values in its coefficients. This is done by obtaining a simple
differential equation of Knizhnik-Zamolodchikov-Bernard-type satisfied by
generating functions of such integrals, and solving it via Picard iteration.
The initial condition involves the generating functions at the cusp $\tau\to
i\infty$ and can be reduced to genus-zero integrals.
