We report the results of a 1-yr Swift X-ray/ultraviolet (UV)/optical programme monitoring the dwarf Seyfert nucleus in NGC 4395 in 2008–2009. The UV/optical flux from the nucleus was found to vary dramatically over the monitoring period, with a similar pattern of variation in each of the observed UV/optical bands (spanning 1900–5500 Å). In particular, the luminosity of NGC 4395 in the 1900 Å band changed by more than a factor of 8 over the monitoring period. The fractional variability was smaller in the UV/optical bands than that seen in the X-rays, with the X-ray/optical ratio increasing with increasing flux. Pseudo-instantaneous flux measurements in the X-ray and each UV/optical band were well correlated, with cross-correlation coefficients of ≥0.7, significant at 99.9 per cent confidence. Archival Swift observations from 2006 sample the intra-day X-ray/optical variability on NGC 4395. These archival data show a very strong correlation between the X-ray and b bands, with a cross-correlation coefficient of 0.84 (significant at >99 per cent confidence). The peak in the cross-correlation function is marginally resolved and asymmetric, suggesting that X-rays lead the b band, but by ≤1 h. In response to recent (2011 August) very high X-ray flux levels from NGC 4395 we triggered Swift target of opportunity observations, which sample the intra-hour X-ray/UV variability. These observations indicate, albeit with large uncertainties, a lag of the 1900 Å band behind the X-ray flux of ∼400 s. The tight correlation between the X-ray and UV/optical lightcurves, together with the constraints we place on the lag time-scale, is consistent with the UV/optical variability of NGC 4395 being primarily due to reprocessing of X-ray photons by the accretion disc.