Rodriguez, who joined from FC Porto for 45 million euros ($61.27 million) in the close season, opened his account with a 19th minute free-kick after having been 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 move up to second on 32 points, two behind leaders Paris St Germain who host Olympique Lyon on Sunday. Lille, third on 30, were playing at struggling neighbours Valenciennes later on Saturday.
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.
($1 = 0.7345 euros)
(Reporting by Gregory Blachier; Editing by Alan Baldwin)
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