  Gravitational waves allow us to test general relativity in the highly
dynamical regime. While current observations have been consistent with waves
emitted by quasi-circular binaries, eccentric binaries may also produce
detectable signals in the near future with ground- and space-based detectors.
We here explore how tests of general relativity scale with the orbital
eccentricity of the source during the inspiral of compact objects up to $e \sim
0.8$. We use a new, 3rd post-Newtonian-accurate, eccentric waveform model for
the inspiral of compact objects, which is fast enough for Bayesian parameter
estimation and model selection, and highly accurate for modeling moderately
eccentric inspirals. We derive and incorporate the eccentric corrections to
this model induced in Brans-Dicke theory and in Einstein-dilaton-Gauss-Bonnet
gravity at leading post-Newtonian order, which suggest a straightforward
eccentric extension of the parameterized post-Einsteinian formalism. We explore
the upper limits that could be set on the coupling parameters of these modified
theories through both a confidence-interval- and Bayes-factor-based approach,
using a Markov-Chain Monte Carlo and a trans-dimensional, reversible-jump,
Markov-Chain Monte Carlo method. We find projected constraints with signals
from sources with $e \sim 0.4$ that are one order of magnitude stronger than
that those obtained with quasi-circular binaries in advanced LIGO. In
particular, eccentric gravitational waves detected at design sensitivity should
be able to constrain the Brans-Dicke coupling parameter $\omega \gtrsim 3300$
and the Gauss-Bonnet coupling parameter $\alpha^{1/2} \lesssim 0.5 \;
{\rm{km}}$ at 90% confidence. Although the projected constraint on $\omega$ is
weaker than other current constraints, the projected constraint on
$\alpha^{1/2}$ is 10 times stronger than the current gravitational wave bound.
