The aperiodic X-ray variability of the 0.7 s X-ray pulsar SMC X-1 is investigated using five EXOSAT observations made in 1984 October and November. During the first observation an X-ray burst was observed, where the source count rate increased by a factor of approximately 3 in less than approximately 0.7 s, and then erratically decayed to the preburst level over approximately 80 s. Following the burst, the intensity level of the persistent emission declined by a factor of approximately 35% over approximately 50 minutes. This decline was associated with an overall hardening of the continuum X-ray spectrum. In contrast, the 1-16 keV spectrum during the burst showed no significant change, and was similar to that of the continuum prior to the burst (a power law with an exponential high-energy cutoff). The shape and relative amplitude of the approximately 0.7 s pulsed signal did not change significantly before, during, and after the burst. This indicates that the burst was not associated with a change in the geometry of the emission region in the vicinity of the neutron star. We consider the possibility that the burst was generated either by a thermonuclear flash of material at the neutron star magnetic pole or by an instability in the accretion flow, and we conclude that the latter is more likely. A power-spectrum analysis of the source light curves reveals the presence of aperiodic flux variations on a time scale from hours to tens of milliseconds. There is a nearly monotonic increase in power toward low frequencies, with an average logarithmic slope of about -1 over the 0.02-1 Hz frequency range. A highly significant peak/turnover at frequencies less than 0.01 Hz is present in the power spectra of the last three observations, and denotes a preferred time scale of a few hundred seconds in the source variability. The burst seems to be an extreme manifestation of the source variability on a time scale of approximately 100 s.