We measure secondary eclipses of the hot giant exoplanets CoRoT-1 at 3.6 and 4.5 μm, and CoRoT-2 at 3.6 μm, both using Warm Spitzer. We find that the Warm Spitzer mission is working very well for exoplanet science. For consistency of our analysis we also re-analyze archival cryogenic Spitzer data for secondary eclipses of CoRoT-2 at 4.5 and 8 μm. We compare the total data for both planets, including optical eclipse measurements by the CoRoT mission, and ground-based eclipse measurements at 2 μm, to existing models. Both planets exhibit stronger eclipses at 4.5 than at 3.6 μm, which is often indicative of an atmospheric temperature inversion. The spectrum of CoRoT-1 is best reproduced by a 2460 K blackbody, due either to a high altitude layer that strongly absorbs stellar irradiance, or an isothermal region in the planetary atmosphere. The spectrum of CoRoT-2 is unusual because the 8 μm contrast is anomalously low. Non-inverted atmospheres could potentially produce the CoRoT-2 spectrum if the planet exhibits line emission from CO at 4.5 μm, caused by tidal-induced mass loss. However, the viability of that hypothesis is questionable because the emitting region cannot be more than about 30% larger than the planet's transit radius, based on the ingress and egress times at eclipse. An alternative possibility to account for the spectrum of CoRoT-2 is an additional opacity source that acts strongly at wavelengths less than 5 μm, heating the upper atmosphere while allowing the deeper atmosphere seen at 8 μm to remain cooler. We obtain a similar result as Gillon et al. for the phase of the secondary eclipse of CoRoT-2, implying an eccentric orbit with e cos(ω) = −0.0030 ± 0.0004.