During moonlit nights, observations with ground-based Cherenkov telescopes at very high energies (VHEs, E > 100 GeV) are constrained since the photomultiplier tubes (PMTs) in the telescope camera are extremely sensitive to the background moonlight. Observations with the VERITAS telescopes in the standard configuration are performed only with a moon illumination less than 35% of full moon. Since 2012, the VERITAS collaboration has implemented a new observing mode under bright moonlight, by either reducing the voltage applied to the PMTs (reduced-high-voltage; RHV configuration), or by utilizing UV-transparent filters. While these operating modes result in lower sensitivity and increased energy thresholds, the extension of the available observing time is useful for monitoring variable sources such as blazars and sources requiring spectral measurements at the highest energies. In this paper we report the detection of gamma-ray flaring activity from the BL Lac object 1ES 1727+502 during RHV observations. This detection represents the first evidence of VHE variability from this blazar. The integral flux is (1.1 +/- 0.2) x 10(-11) cm(-2) s(-1) above 250 GeV, which is about five times higher than the low-flux state. The detection triggered additional VERITAS observations during standard dark-time. Multiwavelength observations with the FLWO 48 '' telescope, and the Swift and Fermi satellites are presented and used to produce the first spectral energy distribution (SED) of this object during gamma-ray flaring activity. The SED is then fitted with a standard synchrotron-self-Compton model, placing constraints on the properties of the emitting region and of the acceleration mechanism at the origin of the relativistic particle population in the jet.