Guly Do Prado was then denied by Bywater before Ramirez found the opener by turning his man and smashing in his first competitive goal for the Saints.
Do Prado hit the post for Southampton as they looked to find a bit of breathing space, but Wednesday were not about to give up, and were only denied an equaliser by a fine Gazzaniga save.
The Premier League side were given a two-goal cushion with just over ten minutes left when referee Mark Halsey pointed to the spot following Joe Mattock's tackle on Steve de Ridder. Rodriguez stepped up and made no mistake despite Bywater getting a hand to it.
Southampton safely saw the match out to book themselves a spot in the League Cup fourth round.