The Argentine made it four goals in two games following his strike against Bordeaux last week, while big-money signing Radamel Falcao struck again.
The hosts, who marked their return to France's top-tier with a 2-0 win over Bordeaux last week, enjoyed a bright start and could have taken the lead as early as the sixth minute, but Lucas Ocampos failed to beat Laurent Pionnier in the opposition goal.
However, boss Claudio Ranieri would not have long to wait to watch his team open the scoring, as striker Radamel Falcao converted from the penalty spot after Ocampos was felled in the area.
Jean Fernandez's men went in search of a swift answer and found one through Victor Hugo Montano, who headed home the equalizer some six minutes later.
As both sides were poised to enter the interval on level terms, Riviere pounced for his first of the game, evading his marker before lobbing the out-rushing Pionnier on the stroke of halftime.
It got worse for the 2012 Ligue 1 champion 12 minutes after the restart, as Yannick Ferreira-Carrasco's corner was met by the boot of Riviere, who side-footed home from close range to establish daylight between the teams.
La Paillade then faced a seemingly insurmountable task after Daniel Congre was dismissed midway through the second half for knocking Ferreira-Carrasco off balance on the edge of the area as the last man.
Riviere completed the rout 10 minutes from time, heading into an empty net after Pionnier could only parry away a deflected shot.
Monaco hosts Toulouse on Friday in a bid to extend its perfect start to the Ligue 1 season, while Montpellier hosts Sochaux on Saturday.