We report on observations of TeV-selected active galactic nuclei (AGNs) made during the first 5.5 months of observations with the Large Area Telescope (LAT) on-board the Fermi Gamma-ray Space Telescope (Fermi). In total, 96 AGNs were selected for study, each being either (1) a source detected at TeV energies (28 sources) or (2) an object that has been studied with TeV instruments and for which an upper limit has been reported (68 objects). The Fermi observations show clear detections of 38 of these TeV-selected objects, of which 21 are joint GeV-TeV sources, and 29 were not in the third EGRET catalog. For each of the 38 Fermi-detected sources, spectra and light curves are presented. Most can be described with a power law of spectral index harder than 2.0, with a spectral break generally required to accommodate the TeV measurements. Based on an extrapolation of the Fermi spectrum, we identify sources, not previously detected at TeV energies, which are promising targets for TeV instruments. Evidence for systematic evolution of the γ-ray spectrum with redshift is presented and discussed in the context of interaction with the extragalactic background light.