The equations of 10 or 11 dimensional supergravity admit supersymmetric compactifications on 7-manifolds of G2 holonomy, but these supergravity vacua are deformed away from special holonomy by the higher-order corrections of string or M-theory. We find simple expressions for the first-order corrections to the Einstein and Killing spinor equations in terms of the calibrating 3-form of the leading-order G2-holonomy background. We thus obtain, and solve explicitly, systems of first-order equations describing the corrected metrics for most of the known classes of cohomogeneity-one 7-metrics with G2 structures.