  Perturbative QCD results in the MSbar scheme can be dramatically improved by
switching to a scheme that accounts for the dominant power law dependence on
the factorization scale in the operator product expansion. We introduce the
``MSR scheme'' which achieves this in a Lorentz and gauge invariant way. The
MSR scheme has a very simple relation to MSbar, and can be easily used to
reanalyze MSbar results. Results in MSR depend on a cutoff parameter R, in
addition to the mu of MSbar. R variations can be used to independently estimate
i) the size of power corrections, and ii) higher order perturbative corrections
(much like mu in MSbar). We give two examples at three-loop order, the ratio of
mass splittings in the B*-B and D*-D systems, and the Ellis-Jaffe sum rule as a
function of momentum transfer Q in deep inelastic scattering. Comparing to
data, the perturbative MSR results work well even for Q ~ 1 GeV, and the size
of power corrections is reduced compared to those in MSbar.
