Context. The young systemsPZ Tel and HD 1160, hosting known low-mass companions, were observed during the commissioning of the new planet finder of the Very Large Telescope (VLT) SPHERE with several imaging and spectroscopic modes. Aims: We aim to refine the physical properties and architecture of both systems. Methods: We use SPHERE commissioning data and dedicated Rapid Eye Mount (REM) observations, as well as literature and unpublished data from VLT/SINFONI, VLT/NaCo, Gemini/NICI, and Keck/NIRC2. Results: We derive new photometry and confirm the short-term (P = 0.94 d) photometric variability of the star PZ Tel A with values of 0.14 and 0.06 mag at optical and near-infrared wavelengths, respectively. We note from the comparison to literature data spanning 38 yr that the star also exhibits a long-term variability trend with a brightening of ~0.25 mag. The 0.63-3.8 μm spectral energy distribution of PZ Tel B (separation ~25 AU) allows us to revise its physical characteristics: spectral type M7 ± 1, Teff = 2700 ± 100 K, log(g) <4.5 dex, luminosity log(L/L⊙) = -2.51 ± 0.10 dex, and mass 38-72 MJ from "hot-start" evolutionary models combining the ranges of the temperature and luminosity estimates. The 1-3.8 μm SED of HD 1160 B (~85 au) suggests a massive brown dwarf or a low-mass star with spectral type M6.0, Teff = 3000 ± 100 K, subsolar metallicity [M/H] = -0.5-0.0 dex, luminosity log(L/L⊙) = -2.81 ± 0.10 dex, and mass 39-166 MJ. The physical properties derived for HD 1160 C (~560 au) from KsL'-band photometry are consistent with the discovery study. The orbital study of PZ Tel B confirms its deceleration and the high eccentricity of its orbit (e > 0.66). For eccentricities below 0.9, the inclination, longitude of the ascending node, and time of periastron passage are well constrained. In particular, both star and companion inclinations are compatible with a system seen edge-on. Based on "hot-start" evolutionary models, we reject other brown dwarf candidates outside 0.25'' for both systems, and giant planet companions outside 0.5'' that are more massive than 3 MJ for the PZ Tel system. We also show that K1-K2 color can be used along with YJH low-resolution spectra to identify young L-type companions, provided high photometric accuracy (≤0.05 mag) is achieved. Conclusions: SPHERE opens new horizons in the study of young brown dwarfs and giant exoplanets using direct imaging thanks to high-contrast imaging capabilities at optical (0.5-0.9 μm) and near-infrared (0.95-2.3 μm) wavelengths, as well as high signal-to-noise spectroscopy in the near-infrared domain (0.95-2.3 μm) from low resolutions (R ~ 30-50) to medium resolutions (R ~ 350). Based on data collected at the European Southern Observatory, Chile, during the commissioning of the SPHERE instrument and ESO programs 085.C-0277, 087.C-0109, 087.C-0535, and 060.A-9026.