We devise a novel resummation prescription based on the method of finite-part integration [Galapon EA. 2017 Proc. R. Soc A 473, 20160567. (doi:10.1098/rspa.2016.0567)] to perform a constrained extrapolation of the divergent weak-field perturbative expansion for the Heisenberg–Euler Lagrangian to the non-perturbative strong magnetic and electric field regimes. In the latter case, the prescription allowed us to reconstruct the non-perturbative imaginary part from a finite collection of the real expansion coefficients. We also demonstrate the utility of the various equivalent representations of Hadamard’s finite part in deriving the exact closed form for the Heisenberg–Euler Lagrangian from the non-perturbative integral representation.