Rodriguez, who joined from FC Porto for 45 million euros in the close season, opened his account with a 19th-minute free kick after being fouled just outside the box.
Striker Anthony Martial then celebrated his first start in the top flight, at the age of 17, by doubling the score from close range one minute before the break.
"I'm really happy but especially with the victory tonight. My goal is important because it helped the team win," playmaker Rodriguez told broadcaster Canal Plus.
Monaco are third on 32 points, two behind leaders Paris St Germain, who host Olympique Lyon on Sunday, and one behind Lille, who won 1-0 at struggling neighbours Valenciennes later on Saturday thanks to a 47th-minute Ronny Rodelin goal.
The northerners managed their 10th consecutive league clean sheet despite being reduced to 10 men eight minutes into the second half, when Franck Beria was shown a second booking for a rough challenge on Valenciennes defender Gary Kagelmacher.
Big-spending Monaco were without joint league top scorer Falcao for the first time after he picked up a thigh injury and was rested.
"Falcao is a very important player but goals always come from collective work," said Rodriguez.
Coach Claudio Ranieri told reporters he will decide on Monday if the Colombian, who has scored nine league goals so far, will make the squad for Tuesday's short trip to Nice.
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