Saints had found themselves on the back foot for much of the opening half-hour at the Madejski Stadium before Rodriguez struck ten minutes before the break.
The striker stretched to poke Rickie Lambert's chipped pass beyond Royals goalkeeper Adam Federici and end the resistance of the Premier League's bottom side.
He then turned provider for Adam Lallana in the second half, dinking the ball over the Reading defence to allow the substitute to wrap up the win.
There was some discontent among the home side with the manner of Rodriguez's goal, though, with there being a suggestion that he had unfairly challenged the home stopper.
But the former Burnley man insisted there was no foul: "My eyes were on the ball so I didn't even see the 'keeper, to be honest.
"Luckily for me, it went in – it wasn't a bad tackle.
"Reading put us under pressure and played really well, but I think the timing of the goal was brilliant for us," he continued.
"It kicked us on a little bit. It was a good game for both teams but we scored some good goals, especially Ads's – that was a great finish.
"I think it was a big win for us today," he continued after seeing his side climb to 11th in the Premier League standings.
"Every game is massive, and now we need to carry on working hard and improving until the end of the season.
"We'll need to take every game as it comes so we're not going to look too much into it (Saints' league position).
"We're going to try to improve the table and hopefully we'll move up even further."
Today's goal was Rodriguez's ninth in Southampton colours, and his third in as many games.
And the 23-year-old says there is plenty more to come from him between now and the end of the season.
"I'm getting a run in the team so I'm getting experience by playing – I think that's what I needed," he reflected.
"I need to get games to improve my form, and now I need to keep working hard because I've got a lot to improve.
"I hope that in each game I can try to get better for the team."