We report the first measurement of the (e,e^{'}p) three-body breakup reaction cross sections in helium-3 (^{3}He) and tritium (^{3}H) at large momentum transfer [⟨Q^{2}⟩≈1.9  (GeV/c)^{2}] and x_{B}>1 kinematics, where the cross section should be sensitive to quasielastic (QE) scattering from single nucleons. The data cover missing momenta 40≤p_{miss}≤500  MeV/c that, in the QE limit with no rescattering, equals the initial momentum of the probed nucleon. The measured cross sections are compared with state-of-the-art ab initio calculations. Overall good agreement, within ±20%, is observed between data and calculations for the full p_{miss} range for ^{3}H and for 100≤p_{miss}≤350  MeV/c for ^{3}He. Including the effects of rescattering of the outgoing nucleon improves agreement with the data at p_{miss}>250  MeV/c and suggests contributions from charge-exchange (SCX) rescattering. The isoscalar sum of ^{3}He plus ^{3}H, which is largely insensitive to SCX, is described by calculations to within the accuracy of the data over the entire p_{miss} range. This validates current models of the ground state of the three-nucleon system up to very high initial nucleon momenta of 500  MeV/c.