We report a combined study of external pressure and Cu-substitution on BaFe 2 As 2 single crystals grown by the in-flux technique. At ambient pressure, the Cu-substitution is known to suppress the spin density wave (SDW) phase in pure BaFe 2 As 2 (T SDW ≈ 140 K) and to induce a superconducting (SC) dome with a maximum transition temperature T max c 4.2 K. This T max c is much lower than the T c ∼ 15-28 K achieved in the case of Ru, Ni and Co substitutions. Such a lower T c is attributed to a Cu 2+ magnetic pair-breaking effect. The latter is strongly suppressed by applied pressure, as shown herein, T c can be significantly enhanced by applying high pressures. In this work, we investigated the pressure effects on Cu 2+ magnetic pair-breaking in the BaFe 2−x Cu x As 2 series. Around the optimal concentration (x opd = 0.11), all samples showed a substantial increase of T c as a function of pressure. Yet for those samples with a slightly higher doping level (over-doped regime), T c presented a dome-like shape with maximum T c 8 K. Remarkably interesting, the under-doped samples, e.g. x = 0.02 display a maximum pressure induced T c 30 K which is comparable to the maximum T c 's found for the pure compound under external pressures. Furthermore, the magnetoresistance effect as a function of pressure in the normal state of the x = 0.02 sample also presented an evolution consistent with the screening of the Cu 2+ local moments. These findings demonstrate that the Cu 2+ magnetic pair-breaking effect is completely suppressed by applying pressure in the low concentration regime of Cu 2+ substituted BaFe 2 As 2 .