The nearby dwarf starburst galaxy NGC 5253 hosts a number of young, massive star clusters, the two youngest of which are centrally concentrated and surrounded by thermal radio emission (the "radio nebula"). To investigate the role of these clusters in the starburst energetics, we combine new and archival Hubble Space Telescope images of NGC 5253 with wavelength coverage from 1500 to 1.9 μm in 13 filters. These include Hα, Pβ, and Pα, and the imaging from the Hubble Treasury Program LEGUS (Legacy Extragalactic UV Survey). The extraordinarily well-sampled spectral energy distributions enable modeling with unprecedented accuracy the ages, masses, and extinctions of the nine optically brightest clusters (MV < -8.8) and the two young radio nebula clusters. The clusters have ages ∼1-15 Myr and masses ∼1 × 104-2.5 × 105 Mo. The clusters' spatial location and ages indicate that star formation has become more concentrated toward the radio nebula over the last ∼15 Myr. The most massive cluster is in the radio nebula; with a mass ∼2.5 × 105 Mo and an age ∼1 Myr, it is 2-4 times less massive and younger than previously estimated. It is within a dust cloud with AV ∼ 50 mag, and shows a clear near-IR excess, likely from hot dust. The second radio nebula cluster is also ∼1 Myr old, confirming the extreme youth of the starburst region. These two clusters account for about half of the ionizing photon rate in the radio nebula, and will eventually supply about 2/3 of the mechanical energy in present-day shocks. Additional sources are required to supply the remaining ionizing radiation, and may include very massive stars.