  The angular momentum flux from an inspiralling binary system of compact
objects moving in quasi-elliptical orbits is computed at the third
post-Newtonian (3PN) order using the multipolar post-Minkowskian wave
generation formalism. The 3PN angular momentum flux involves the instantaneous,
tail, and tail-of-tails contributions as for the 3PN energy flux, and in
addition a contribution due to non-linear memory. We average the angular
momentum flux over the binary's orbit using the 3PN quasi-Keplerian
representation of elliptical orbits. The averaged angular momentum flux
provides the final input needed for gravitational wave phasing of binaries
moving in quasi-elliptical orbits. We obtain the evolution of orbital elements
under 3PN gravitational radiation reaction in the quasi-elliptic case. For
small eccentricities, we give simpler limiting expressions relevant for phasing
up to order $e^2$. This work is important for the construction of templates for
quasi-eccentric binaries, and for the comparison of post-Newtonian results with
the numerical relativity simulations of the plunge and merger of eccentric
binaries.
