Swift discovered GRB 050713A and slewed promptly to begin observing with its narrow-field instruments 72.6 s after the burst onset, while the prompt gamma-ray emission was still detectable in the BAT. Simultaneous emission from two flares is detected in the BAT and XRT. This burst marks just the second time that the BAT and XRT have simultaneously detected emission from a burst and the first time that both instruments have produced a well-sampled, simultaneous data set covering multiple X-ray flares. The temporal rise and decay parameters of the flares are consistent with the internal-shock mechanism. In addition to the Swift coverage of GRB 050713A, we report on the Konus-Wind (K-W) detection of the prompt emission, an upper limiting GeV measurement of the prompt emission made by the MAGIC imaging atmospheric Cerenkov telescope, and XMM-Newton observations of the afterglow. Simultaneous observations with Swift XRT and XMM-Newton produce consistent results, showing a break in the light curve at T0+~15 ks. Together, these four observatories provide unusually broad spectral coverage of the prompt emission and detailed X-ray follow-up of the afterglow for 2 weeks after the burst trigger. Simultaneous spectral fits of K-W with BAT and BAT with XRT data indicate that an absorbed broken power law is often a better fit to GRB flares than a simple absorbed power law. These spectral results together with the rapid temporal rise and decay of the flares suggest that flares are produced in internal shocks due to late-time central-engine activity.