We give a general surgery formula for the Casson-Walker-Lescop invariant of closed 3-manifolds, by regarding this invariant as the leading term of the LMO invariant. Our proof is diagrammatic and combinatorial, and provides a new viewpoint on a formula established by C. Lescop for her extension of the Walker invariant. A central ingredient in our proof is an explicit identification of the coefficients of the Conway polynomial as combinations of coefficients in the Kontsevich integral. This latter result relies on general 'factorization formulas' for the Kontsevich integral coefficients.