  Many studies have recently documented the orbital response of eccentric
binaries accreting from thin circumbinary disks, characterizing the change in
binary semi-major axis and eccentricity. We extend these calculations to
include the precession of the binary's longitude of periapse induced by the
circumbinary disk, and we characterize this precession continuously with binary
eccentricity $e_b$ for equal mass components. This disk-induced apsidal
precession is prograde with a weak dependence on binary eccentricity when $e_b
\lesssim 0.4$ and decreases approximately linearly for $e_b \gtrsim 0.4$; yet
at all $e_b$ binary precession is faster than the rates of change to the
semi-major axis and eccentricity by an order of magnitude. We estimate that
such precession effects are likely most important for sub-parsec separated
binaries with masses $\lesssim 10^7 M_\odot$, like LISA precursors. We find
that accreting, equal-mass LISA binaries with $M < 10^6 M_\odot$ (and the most
massive $M \sim 10^7 M_\odot$ binaries out to $z \sim 3$) may acquire a
detectable phase offset due to the disk-induced precession. Moreover,
disk-induced precession can compete with General Relativistic precession in
vacuum, making it important for observer-dependent electromagnetic searches for
accreting massive binaries -- like Doppler boost and binary self-lensing models
-- after potentially only a few orbital periods.
