Superluminous supernovae (SLSNe) radiate ≳10–100 times more energy than ordinary stellar explosions, implicating a novel power source behind these enigmatic events. One frequently discussed source, particularly for hydrogen-poor (Type I) SLSNe, is a central engine such as a millisecond magnetar or accreting black hole. Both black hole and magnetar engines are expected to channel a fraction of their luminosity into a collimated relativistic jet. Using 3D relativistic hydrodynamical simulations, we explore the interaction of a relativistic jet, endowed with a luminosity L j ≈ 1045.5 erg s−1 and duration t eng ≈ 10 days compatible with those needed to power SLSNe, launched into the envelope of the exploding star. The jet successfully breaks through the expanding ejecta, and its shocked cocoon powers ultraviolet/optical emission lasting several days after the explosion and reaching a peak luminosity ≳1044 erg s−1, corresponding to a sizable fraction of L j . This high radiative efficiency is the result of the modest adiabatic losses the cocoon experiences owing to the low optical depths of the enlarged ejecta at these late times, e.g., compared to the more compact stars in gamma-ray bursts. The luminosity and temperature of the cocoon emission match those of the “bumps” in SLSN light curves observed weeks prior to the optical maximum in many SLSNe. Confirmation of jet breakout signatures by future observations (e.g., days-long to weeks-long internal X-ray emission from the jet for on-axis observers, spectroscopy confirming large photosphere velocities v/c ≳ 0.1, or detection of a radio afterglow) would offer strong evidence for central engines powering SLSNe.