Spectroscopic eclipsing binaries (SEBs) are fundamental benchmarks in stellar astrophysics and today are observed in breathtaking detail by missions like the Transiting Exoplanet Survey Satellite (TESS), Kepler, and Apache Point Observatory Galactic Evolution Experiment (APOGEE). We develop a methodology for simultaneous analysis of high-precision Kepler light curves and high-resolution near-infrared spectra from APOGEE and present orbital solutions and evolutionary histories for a subset of SEBs within this overlap. Radial velocities extracted from APOGEE spectra using the broadening function (BF) technique are combined with Kepler light curves and to yield binary orbital solutions. The BF approach yields more precise radial velocities than the standard cross-correlation function, which in turn yields more precise orbital parameters and enables the identification of tertiary stars. The orbital periods of these seven SEBs range from 4 to 40 days. Four of the systems (KIC 5285607, KIC 6864859, KIC 6778289, and KIC 4285087) are well-detached binaries. The remaining three systems have apparent tertiary companions, but each exhibit two eclipses along with at least one spectroscopically varying component (KIC 6449358, KIC 6131659, and KIC 6781535). Gaia distances are available for four targets which we use to estimate temperatures of both members of these SEBs. We explore evolutionary histories in H-R diagram space and estimate ages for this subset of our sample. Finally, we consider the implications for the formation pathways of close binary systems via interactions with tertiary companions. Our methodology combined with the era of big data and observation overlap opens up the possibility of discovering and analyzing large numbers of diverse SEBs, including those with high flux ratios and those in triple systems.