The two goals were Rodriguez's first goals with Southampton in official competition, as the young striker took advantage of playing as a central striker instead of on the wing as Nigel Adkins had deployed him mostly in previous matches.
Southampton will learn their opponent in the next round when the draw is conducted Wednesday night after the completion of the Wednesday fixtures, around 10 p.m. BST.
"Reaching the last 16 is a great achievement but we still want to go further, so we'll take any team and show them how good we are so we can hopefully get through to the next stage," Rodriguez told the Southampton website after the match.
Paulo Gazzaniga got his second straight start in the Southampton net with Kelvin Davis on the bench. Artur Boruc was left out of the squad after signing a one-year deal with the Saints on Saturday.
Maya Yoshida was the only other Southampton player to remain in the starting lineup as Adkins chose to get some other members of the squad some work. Jos Hooiveld returned from his calf injury that he suffered against Arsenal, and Jack Cork made his first appearance in the senior squad after recovering from an ankle injury suffered in a preseason friendly against Bristol City.
Rodriguez got Southampton on the board in the 30th minute when he collected a pass from Richard Chaplow at the top of the box, turned sharply past a Sheffield Wednesday defender and blasted a shot past Sheffield Wednesday goalkeeper Stephen Bywater. Saints would retain the 1-0 lead into halftime.
The second goal was set up by Steve De Ridder, who was making his first appearance of the season. De Ridder made a pacey run down the left flank into the penalty area, where Joe Mattock clumsily brought down the Belgian winger.
Rodriguez stepped to the spot in the 78th minute to take the penalty, and found himself a bit fortunate as Bywater got a good piece of the penalty, but the ball trickled over the line to give Southampton a comfortable 2-0 lead they would hold the rest of the way.
"I just want to play, and to get two goals for the team is brilliant. I'm really chuffed," Rodriguez told the Southampton website after the match.
There was a late injury worry as Cork was forced off the pitch in the 80th minute. The positive side of the injury was that it allowed Saints Academy product Andy Robinson to make his senior team debut.
Southampton will hope to keep their good run of form going in a tricky away fixture Saturday as they travel to Goodison Park to take on Everton.