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Abstract Modular graph functions arise in the calculation of the low-energy expansion of closed-string scattering amplitudes. For toroidal world-sheets, they are SL(2, ℤ)-invariant functions of the torus complex structure that have to be integrated over the moduli space of inequivalent tori. We use methods from resurgent analysis to construct the non-perturbative corrections arising for two-loop modular graph functions when the argument of the function approaches the cusp on this moduli space. SL(2, ℤ)-invariance will in turn strongly constrain the behaviour of the non-perturbative sector when expanded at the origin of the moduli space.