Rodriguez, who delivered another brilliant performance in a playmaking position, put Monaco ahead on the stroke of halftime with a superb free-kick.
He completed his first double for the side in the 77th minute following a lightening counter-attack. Monaco moved up to 53 points in second place.
PSG have 58 after they defeated lowly Valenciennes 3-0 at home on Friday.
"We produced a great game, we played well and I scored two goals so it's a perfect night," Rodriguez told broadcaster Canal Plus.
The Colombia international, who has now scored five goals and set up nine others, joined Monaco from Porto for 45 million euros ($61.59 million) last summer.
Emmanuel Riviere created the first chance for the visitors after seven minutes but his acrobatic effortt was parried away by goalkeeper Jean-Louis Leca.
A Layvin Kurzawa header from a deep Joao Moutinho cross narrowly missed the target and Monaco seemed unable to threaten Leca again until Rodriguez was fouled just outside the box in the 45th minute.
He took the free kick and powerfully bent the ball in the net.
The talented Colombian initiated some more good moves after the break but his partners lacked a cutting-edge.
He again did the job himself with 13 minutes left when he grabbed the ball in midfield and ran with it and seal Monaco's seventh away victory.
In Saturday's later matches, Stade Reims continued their pursuit of European football with a 1-0 win over Bordeaux while Toulouse beat Lorient 3-1 to leapfrog their opponents in the table.
Sochaux boosted their survival hopes with a 1-0 win over Guingamp, Rennes and Montpellier shared four goals to take a point apiece, and Nice and Nantes played out a goalless draw.