Joseph was red-carded by referee Andy D'Urso for elbowing Bojan Djordjic, but disciplined Hull never felt his absence and, once Elliott scored, there was only going to be one winner.
Even the return of Taribo West failed to inspire Argyle, who have now picked up just one point from three home games after an opening-day victory at Reading.
Nigerian international West was stranded out of position upfield when Craig Fagan broke free and fed Elliott, after whose goal Hull staged a magnificent rearguard action to preserve the victory.
Argyle, who had only two fit senior strikers following Nick Chadwick's ankle injury, were forced to give a Championship debut to substitute Chris Zebroski, but he, like his colleagues, could make no headway.
Nine, sometimes all ten, of Hull's men stayed resolutely behind the ball and Argyle found it impossible to break them down.