  We study generating functions of moduli-space integrals at genus one that are
expected to form a basis for massless $n$-point one-loop amplitudes of open
superstrings and open bosonic strings. These integrals are shown to satisfy the
same type of linear and homogeneous first-order differential equation w.r.t.
the modular parameter $\tau$ which is known from the A-elliptic
Knizhnik--Zamolodchikov--Bernard associator. The expressions for their
$\tau$-derivatives take a universal form for the integration cycles in planar
and non-planar one-loop open-string amplitudes. These differential equations
manifest the uniformly transcendental appearance of iterated integrals over
holomorphic Eisenstein series in the low-energy expansion w.r.t. the inverse
string tension $\alpha'$. In fact, we are led to matrix representations of
certain derivations dual to Eisenstein series. Like this, also the
$\alpha'$-expansion of non-planar integrals is manifestly expressible in terms
of iterated Eisenstein integrals without referring to twisted elliptic multiple
zeta values. The degeneration of the moduli-space integrals at $\tau
\rightarrow i\infty$ is expressed in terms of their genus-zero analogues --
$(n{+}2)$-point Parke--Taylor integrals over disk boundaries. Our results yield
a compact formula for $\alpha'$-expansions of $n$-point integrals over
boundaries of cylinder- or Moebius-strip worldsheets, where any desired order
is accessible from elementary operations.
