Let be simply connected, complete, with non-positive sectional curvatures, and a 2-dimensional closed integral current (or flat chain mod 2) with compact support in . Let be an area minimising integral 3-current (resp.~flat chain mod 2) such that . We use a weak mean curvature flow, obtained via elliptic regularisation, starting from , to show that S satisfies the optimal Euclidean isoperimetric inequality: . We also obtain an optimal estimate in case the sectional curvatures of M are bounded from above by and characterise the case of equality. The proof follows from an almost monotonicity of a suitable isoperimetric difference along the approximating flows in one dimension higher and an optimal estimate for the Willmore energy of a 2-dimensional integral varifold with first variation summable in .