  We introduce a new paradigm for constructing accurate analytic waveforms (and
fluxes) for eccentric compact binaries. Our recipe builds on the standard
Post-Newtonian (PN) approach but (i) retains implicit time-derivatives of the
phase space variables in the instantaneous part of the noncircular waveform,
and then (ii) suitably factorizes and resums this partly PN-implicit waveform
using effective-one-body (EOB) procedures. We test our prescription against the
exact results obtained by solving the Teukolsky equation with a test-mass
source orbiting a Kerr black hole, and compare the use of the exact vs PN
equations of motion for the time derivatives computation. Focusing only on the
quadrupole contribution, we find that the use of the exact equations of motion
yields an analytical/numerical agreement of the (averaged) angular momentum
fluxes that is improved by $40\%$ with respect to previous work, with $4.5\%$
fractional difference for eccentricity $e=0.9$ and black hole dimensionless
spin $-0.9\leq \hat{a}\leq +0.9$. We also find a remarkable convergence trend
between Newtonian, 1PN and 2PN results. Our approach carries over to the
comparable mass case using the resummed EOB equations of motion and paves the
way to faithful EOB-based waveform model for long-inspiral eccentric binaries
for current and future gravitational wave detectors.
