We report three additional Spectrograph for Integral Field Observations in the Near Infrared (SINFONI) detections of Hα emission line from quasar absorbers, two of which are new identifications. These were targeted among a sample of systems with and metallicities measured from high-resolution spectroscopy. The detected galaxies are at impact parameters ranging from 6 to 12 kpc from the quasar’s line of sight. We derive star formation rates (SFRs) of a few M⊙ yr−1 for the two absorbers at zabs∼ 1 and SFR = 17 M⊙ yr−1 for the damped Lyman α system (DLA) at zabs∼ 2. These three detections are found among a sample of 16 DLAs and sub-DLAs (five at zabs∼ 1 and seven at zabs∼ 2). For the remaining undetected galaxies, we derive flux limits corresponding to SFR < 0.1–11.0 M⊙ yr−1 depending on redshift of the absorber and depth of the data. When combined with previous results from our survey for galaxy counterparts to H i-selected absorbers, we find a higher probability of detecting systems with higher metallicity as traced by dust-free [Zn/H] metallicity. We also report a higher detection rate with SINFONI for host galaxies at zabs∼ 1 than for systems at zabs∼ 2. Using the [N ii]/Hα ratio, we can thus compare absorption and emission metallicities in the same high-redshift objects, more than doubling the number of systems for which such measures are possible.