Low-ionization transitions such as the MgII 2796/2803 doublet trace cold gas in the vicinity of galaxies. It is not clear whether this gas is part of the interstellar medium of large proto-disks, part of dwarfs, or part of entrained material in supernovae-driven outflows. Studies based on MgII statistics, e.g. stacked images and clustering analysis, have invoked starburst-driven outflows where MgII absorbers are tracing the denser and colder gas of the outflow. A consequence of the outflow scenario is that the strongest absorbers ought to be associated with starbursts. We use the near-IR integral field spectrograph SINFONI to test whether starbursts are found around z~1 MgII absorbers. For 67% (14 out of 21) of the absorbers with rest-frame equivalent width larger than 2 AA, we do detect Ha in emission within 200 km/s of the predicted wavelength based on the \MgII redshift. The star-formation rate (SFR) inferred from Halpha ranges from 1 to 20 Msun/yr, i.e. showing a level of star-formation larger than in M82 by a factor of >4 on average. Our flux limit (3-sigma) corresponds to a SFR of 0.5 Msun/yr. We find evidence (at >95% confidence) for a correlation between SFR and equivalent width, indicating a physical connection between starburst phenomena and gas seen in absorption. In the cases where we can extract the velocity field, the host-galaxies reside in halos with mean mass <log M_h>=11.2 in good agreement with clustering measurements.