The youngest super star clusters (SSCs) in the merging Antennae Galaxies (NGC 4038/39) drive supersonic mass-loaded outflows from the HII regions in which they are embedded. High-resolution K-band Keck/NIRSPEC spectroscopy reveals broad, spatially extended Brγ emission in 16 targets. Simple wind models for the line profiles provide good fits and imply cluster mass-loss rates of up to 1.5 M⊙/year and terminal velocities of up to 205 km s−1. The emission-line clusters (ELCs) that drive these outflows constitute at least 15% of the star formation rate in the Antennae, and their high star formation efficiencies imply that they will probably evolve into bound SSCs. The youngest ELC outflows, which are driven primarily by stellar winds, very efficiently entrain ambient matter. The cluster winds transfer or dissipate a large fraction of their energy and momentum in a hot or cool medium that does not emit Brγ. ELCs are the individual engines that power galactic-scale superwinds, viewed in their earliest evolutionary stage.