We report an electrochemically assisted jump-to-contact scanning tunneling microscopy (STM) break junction approach to create reproducible and well-defined single-molecule spintronic junctions. The STM break junction is equipped with an external magnetic field either parallel or perpendicular to the electron transport direction. The conductance of Fe-terephthalic acid (TPA)-Fe single-molecule junctions is measured and a giant single-molecule tunneling anisotropic magnetoresistance (T-AMR) up to 53% is observed at room temperature. Theoretical calculations based on first-principles quantum simulations show that the observed AMR of Fe-TPA-Fe junctions originates from electronic coupling at the TPA-Fe interfaces modified by the magnetic orientation of the Fe electrodes with respect to the direction of current flow. The present study highlights new opportunities for obtaining detailed understanding of mechanisms of charge and spin transport in molecular junctions and the role of interfaces in determining the MR of single-molecule junctions.