Porphyrins and related molecules with strong two-photon absorption (TPA) are extremely called for because of several emerging applications, including 3D optical memory, high-resolution fluorescence microscopy and photodynamic therapy. In this paper we demonstrate for the first time that an asymmetric meso-substitution of porphyrin macrocycle with electron-donating diphenylamino-stilbene or bis-(diphenylamino)-stilbene groups results in a drastic enhancement of intrinsic TPA cross section in the near-IR region. The cross section value amounts to 500 - 900 GM depending on substituent group and link structure, which is about 102 times the corresponding value for the unsubstituted parent molecule. Compared to symmetrical porphyrins, the TPA spectra of this series follow qualitatively the corresponding one-photon spectra. Therefore, we describe the observed TPA spectra and absolute cross section values by taking into account the change of permanent dipole moment upon excitation. A new zeroth-generation dendrimer, consisting of a porphyrin core, symmetrically tertakis-meso-substituted with strong TPA dendrons, reveals 7 times increase of the cross section (740 vs 110 GM) as compared to its mono-meso-substituted analogue. We also demonstrated an efficient singlet oxygen generation upon two-photon excitation of these new molecules, which makes them particularly attractive for photodynamic therapy.