We have carried out the electrical resistivity and ac specific heat measurements under pressure P and magnetic field H in a pressure-induced superconductor CeIrSi3 with the noncentrosymmetric tetragonal crystal structure. Superconductivity is observed in a wide pressure region from 1.9 GPa to about 3.5 GPa. The antiferromagnetic state disappears at P-c = 2.25 GPa under H = 0, but in magnetic field we have found an antiferromagnetic state. The corresponding electrical resistivity in constant magnetic field decreases rather steeply below the Neel temperature T-N under pressures up to about 2.4 GPa, while it shows an upturn anomaly at T-N under pressures larger than 2.4 GPa. The ac specific heat also indicates a clear jump at T-N in magnetic field. When the antiferromagnetic phase exists in magnetic field, the superconducting transition becomes broad, possessing a transition width Delta T-sc = 0.2-0.5 K. On the other hand, it becomes extremely sharp, with Delta T-sc = 0.03K even at 17 T under P = 2.55 GPa, for example, when the antiferromagnetic phase disappears. The upper critical field H-c2 for H parallel to [001] is also extremely enhanced around 2.5-2.7 GPa, and the H-c2(0) value reaches about 45 T at P-c* similar or equal to 2.63 GPa, even though the superconducting transition temperature T-sc = 1.6K is very small. The electronic instability is realized in magnetic field at P-c*. The present characteristic heavy fermion superconductivity in the noncentrosymmetric structure is compared with the similar pressure-induced superconductivity of CeRhIn5.