The transverse and longitudinal magnetoelectric susceptibilities (MES) were quantitatively determined for (001) heteroepitaxial BiFeO$_{3}$-CoFe$_{2}$O$_{4}$ nanostructures. Both of these MES values were sharply enhanced at magnetic fields below 6 kOe and revealed asymmetric lineshapes with respect to the dc magnetic field, demonstrating the strain-induced magnetoelectric effect. The maximum transverse MES, which reached as high as $∼$60 mV/cm Oe, was about five times larger than the longitudinal MES. This observation signifies that transverse magnetostriction of the CoFe$_{2}$O$_{4}$ nanopillars is enhanced more than the bulk value due to preferred magnetic domain alignment along the [001] direction coming from compressive, heteroepitaxial strain. ; Comment: 8 pages, 3 figures