Giant magnetostriction (GMS) reported until now in exchange-coupled multilayers (ECML) present quasi-isotropic magnetostrictive behavior. In the present work it is shown the possibility to induce highly anisotropic GMS at low field with the single domain state in (TbFe/Fe)n multilayers when deposited by ion sputtering under a magnetic polarization field Hd. ECML grown under bias field Hd, applied along the large size of rectangular glass substrates present original magnetic and magnetoelastic (ME) properties associated with strong uniaxial behavior. After saturation by an external field H∥ applied parallel to the easy axis induced by Hd, a single domain state is kept by decreasing H∥ down to zero (Mr/Ms=1) which explains why there is no change of the magnetoelastic coefficient bγ,2 by increasing or decreasing H∥. A different behavior arises along the hard magnetization axis with a strong change of bγ,2 by increasing H∥ up to the saturation. ME anisotropy is the origin of high amplitude flexural and torsional oscillation modes as observed in our ECML.