The (d) over bar - (u) over bar flavor asymmetry in the proton arising from pion loops is computed using chiral effective field theory. The calculation includes both nucleon and Delta intermediate states, and uses both the fully relativistic and heavy baryon frameworks. The x dependence of extracted from the Fermilab E866 Drell-Yan data can be well reproduced in terms of a single transverse momentum cutoff parameter regulating the ultraviolet behavior of the loop integrals. In addition to the distribution at x > 0, corrections to the integrated asymmetry from zero momentum contributions are computed, which arise from pion rainbow and bubble diagrams at x = 0. These have not been accounted for in previous analyses, and can make important contributions to the lowest moment of (d) over bar - (u) over bar.