  We present a systematic phase curve analysis of known transiting systems
observed by the Transiting Exoplanet Survey Satellite (TESS) during year one of
the primary mission. Using theoretical predictions for the amplitude of the
planetary longitudinal atmospheric brightness modulation, stellar ellipsoidal
distortion and Doppler boosting, as well as brightness considerations to select
targets with likely detectable signals, we applied a uniform data processing
and light-curve modeling framework to fit the full-orbit phase curves of 22
transiting systems with planet-mass or brown dwarf companions, including
previously published systems. Statistically significant secondary eclipse
depths and/or atmospheric brightness modulation amplitudes were measured for
HIP 65A, WASP-18, WASP-19, WASP-72, WASP-100, WASP-111, WASP-121, and
WASP-122/KELT-14. For WASP-100b, we found marginal evidence that the brightest
region of the atmosphere is shifted eastward away from the substellar point. We
detected significant ellipsoidal distortion signals in the light curves of HIP
65A, TOI-503, WASP-18, and WASP-30, with HIP 65A, TOI-503 and WASP-18 also
exhibiting Doppler boosting. The measured amplitudes of these signals agree
with the predictions of theoretical models. Combining the optical secondary
eclipse depths with previously published Spitzer 3.6 and 4.5 $\mu$m
measurements, we derived dayside brightness temperatures and visible-light
geometric albedos for a subset of the analyzed systems. We also calculated
updated transit ephemerides combining the transit timings from the TESS light
curves with previous literature values.
