Claudio Ranieri's side went into the game knowing defeat would give Laurent Blanc's men the chance to wrap up the title with a win next week.
And with injured striker Radamel Falcao watching from the stands, Monaco produced a show of defiance to delay the destination of the title.
Rodriguez scored after 18 minutes and Andrea Raggi doubled the lead in the 72nd minute.
A spot kick from Rodriguez made the lead more emphatic and although Alejandro Bedoya grabbed a consolation, Monaco easily wrapped up the win to bounce back impressively after last week's defeat to Evian.
Fourth-placed Saint-Etienne lost ground in the race for third after drawing 1-1 with mid-table Nice.
Mathieu Bodmer headed the visitors in front after 57 minutes but the hosts grabbed a late equaliser from the penalty spot after Didier Digard's injury-time handball which saw him receive his second yellow card.
Max Gradel converted the spot kick but they are still five points from third.
Lyon kept their faint UEFA Champions League qualification hopes alive with a 2-1 win at 10-man Valenciennes who moved closer to relegation.
Bafetimbi Gomis bagged his 12th league goal of the campaign on the half hour but Abdul Waris equalised midway through the second half.
Parity was shortlived though, as Jordan Ferri struck what turned out to be the winner after 69 minutes.
Lyon's unused substitute goalkeeper Remy Vercoutre was shown two yellow cards for dissent late on, while Marco da Silva was dismissed for Valenciennes after a second booking.
Waris thought he had equalised two minutes later but it was disallowed and Lyon held on to remain nine points behind Lille in third, while Valenciennes are now six points from safety.
