Using UV-to-H broad and narrow-band HST imaging, we derive the ages and masses of the 11 brightest star clusters in the dwarf galaxy NGC5253. This galaxy, located at ~3 Mpc, hosts an intense starburst, which includes a centrally-concentrated dusty region with strong thermal radio emission (the `radio nebula'). The HST imaging includes data from the Cycle 21 Treasury Program LEGUS (Legacy ExtraGalactic UV Survey), in addition to narrow--band H-alpha (6563 A), P-beta (12820 A), and P-alpha (18756 A). The bright clusters have ages ~1-15 Myr and masses ~1E4 - 2.5E5 Msun. Two of the 11 star clusters are located within the radio nebula, and suffer from significant dust attenuation. Both are extremely young, with a best-fit age around 1 Myr, and masses ~7.5E4 and ~2.5E5 Msun, respectively. The most massive of the two `radio nebula' clusters is 2-4 times less massive than previously estimated and is embedded within a cloud of dust with A_V~50 mag. The 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.