The £6million summer acquisition from Burnley impressed in front of the 9,890 spectators at St Mary’s on Tuesday night as the hosts reach the last 16 for the second successive season.
After earlier being denied in a one-on-one with goalkeeper Stephen Bywater, Rodriguez broke the deadlock in the 30th minute by turning brilliantly in the box before rifling home.
Guly do Prado hit the post as Nigel Adkins’ side pushed for a second that came 12 minutes from time after referee Mark Halsey adjudged substitute Joe Mattock to have fouled Steve de Ridder in the box.
Bywater got to Rodriguez’s penalty but it trickled home as Southampton comfortably made it two wins in four days, while Dave Jones saw his former club condemn Wednesday to a fifth straight defeat in all competitions.
Coming into the match both sides made it abundantly clear where their priorities lay, with the Owls making seven changes to their starting line-up and the hosts nine.
Only Paulo Gazzaniga and Maya Yoshida remained from the side that secured Saints’ first Barclays Premier League win of the campaign against Aston Villa on Saturday, while Adkins was able to name Jack Cork in midfield for the first time since a pre-season ankle injury.
The hosts started brightly and De Ridder saw a through-ball to Rodriguez snuffed out by Bywater inside the opening minute.
Do Prado fired a half-volley wide shortly afterwards, before the south Yorkshire outfit began to threaten.
Former Southampton loanee Michail Antonio was looking particularly dangerous for the Owls, who saw Mark Beevers nod wide in the 10th minute.
Rodriguez was denied one-on-one by Bywater as play began to swing from end to end, with defensive confusion almost seeing Antonio net at the other end shortly afterwards.
The winger again came close as he beat Gazzaniga to a ball only to chip onto the roof of the net, although the hosts were beginning to find their feet.
After hooking a Richard Chaplow cross wide, Do Prado forced Bywater into a fine save before Frazer Richardson’s cross-cum-shot threatened to dip in.
Southampton’s continued pressure translated into the opening goal on the half-hour mark when Rodriguez collected a ball from Chaplow, before turning his markers and lashing home.
A dominant spell for the hosts followed and eight minutes later only the post prevented Do Prado from doubling their lead with a header.
Miguel Llera almost turned into his own net when the ball was played back into the box but Wednesday held firm to ensure the deficit was only one goal at the break.
Mattock replaced Kieran Lee as the visitors began the second period well, with Nejc Pecnik coming close with an audacious effort six minutes in.
Wednesday were enjoying a good spell of pressure as Pecnik again attempted to equalise but, despite working space in the Saints box, could not find a way past Gazzaniga.
Llera received the first booking of the evening for a challenge on Do Prado as Saints returned to the attack, with captain Jos Hooiveld – returning on Tuesday night from a calf injury – directing a header straight at Bywater from the resulting free-kick.
Wednesday soon brought on Jermaine Johnson and Chris O’Grady in a bid to change the course of the match, before Saints made a move of their own by replacing Do Prado with Ben Reeves.
The substitute was in the heart of the action immediately and saw a shot blocked, before blazing over unbalanced from the edge of the box.
Reeves’ introduction brought a new urgency to Southampton, who were awarded a 78th-minute penalty by referee Halsey after De Ridder was brought down by Mattock.
Despite the Owls’ insistence that the foul was made outside the area, Rodriguez stepped up and, although Bywater got to it, the ball squirmed in.
It capped a successful evening for Southampton in which the only negative was a knock to Cork nine minutes from time.
Andy Robinson replaced him for his debut and Sam Hoskins followed him on shortly afterwards, before Johnson was twice denied a consolation goal by Gazzaniga.