Rodriguez left Porto for a transfer fee of 45 million euros in the close season but had only started two previous games for Monaco.
Yannick Ferreira-Carrasco collected the Colombian forward's defence-splitting pass in the 15th minute before coolly slotting the ball past keeper Stephane Ruffier.
"It was easy for me, I just had to poke the ball home. It was an extraordinary pass," Ferreira-Carrasco told Canal Plus.
Monaco captain Eric Abidal said: "We knew he (Rodriguez) was a technically-gifted player."
Romain Hamouma levelled for St Etienne four minutes into the second half with a low cross-shot.
Lucas Ocampos then grabbed the winner for Monaco three minutes later, keeping his composure to convert a fine cross from Rodriguez.
Rodriguez also went close to a stoppage-time goal when Ruffier palmed away the 22-year-old's long-range free kick.
St Etienne, seventh with 13 points from nine matches, have now lost three of their last four home games.
"We lack character, we need to question ourselves. Our first half was terrible," said Hamouma.
Paris St Germain, second on 18 points, travel to arch-rivals Olympique Marseille on Sunday minus injured centre back Thiago Silva and forward Jeremy Menez who has been left out for disciplinary reasons.
Marseille are third with 17 points from eight games.