We present simultaneous X-ray and radio observations of the black hole X-ray binary V404 Cygni at the end of its 2015 outburst. From 2015 July 11-August 5 we monitored V404 Cygni with Chandra, Swift, and NuSTAR in the X-ray, and with the Karl G. Jansky Very Large Array and the Very Long Baseline Array in the radio, spanning a range of luminosities that were poorly covered during its previous outburst in 1989 (our 2015 campaign covers 2e33<Lx<1e34 erg/s). During our 2015 campaign, the X-ray spectrum evolved rapidly from a hard photon index of Gamma~1.6 (at Lx~1e34 erg/s) to a softer Gamma~2 (at Lx~3e33 erg/s). We argue that V404 Cygni reaching Gamma~2 marks the beginning of the quiescent spectral state, which occurs at a factor of ~3-4 higher X-ray luminosity than the average pre-outburst luminosity of ~8e32 erg/s. V404 Cygni falls along the same radio/X-ray luminosity correlation that it followed during its previous outburst in 1989, implying a robust disk-jet coupling. We exclude the possibility that a synchrotron cooled jet dominates the X-ray emission in quiescence, leaving synchrotron self-Compton from either a hot accretion flow or from a radiatively cooled jet as the most likely sources of X-ray radiation, and/or particle acceleration along the jet becoming less efficient in quiescence. Finally, we present the first indications of correlated radio and X-ray variability on minute timescales in quiescence, tentatively measuring the radio emission to lag the X-ray by 15+/-4 min, suggestive of X-ray variations propagating down a jet of length <3.0 AU.