We report on the first axion search results from the Cryogenic Dark Matter Search (CDMS) experiment at the Soudan Underground Laboratory. An energy threshold of 2 keV for electron-recoil events allows a search for possible solar axion conversion into photons or local galactic axion conversion into electrons in the germanium crystal detectors. The solar axion search sets an upper limit on the Primakov coupling g(agammagamma) of 2.4x10(-9) GeV-1 at the 95% confidence level for an axion mass less than 0.1 keV/c2. This limit benefits from the first precise measurement of the absolute crystal plane orientations in this type of experiment. The galactic axion search analysis sets a world-leading experimental upper limit on the axioelectric coupling g(aee) of 1.4x10(-12) at the 90% confidence level for an axion mass of 2.5 keV/c2.