After a scrappy opening spell from both sides, Adam Le Fondre found himself in a good position to put the home side ahead, but he did not get a clean contact with Danny Guthrie's free kick.
The Royals appealed for a penalty moments later, believing that Hal Robson-Kanu was brought down in the box, but the referee waved it away.
Reading's dominance continued as Adrian Mariappa tried to head the ball in, but Artur Boruc was on hand to make the save for the Saints.
However, it was the visitors who broke the deadlock in the 35th minute, as Rodriguez and Ricky Lambert teamed up, with Rodriguez beating Adam Federici to the ball, sliding it into the net to put Mauricio Pochettino's side ahead.
The 23-year-old could have added further goals, having struck the bar two minutes after his goal before firing the ball wide just before half time.