Rodriguez, who left Turf Moor for the south-coast club in 2012, came off the bench to put Southampton 3-2 up in the 66th minute after second-half strikes from Sam Vokes and Danny Ings had cancelled out goals from Nathaniel Clyne and Rickie Lambert in the opening period.
Another substitute, Adam Lallana, fired in a long-range effort at St Mary's Stadium to put Southampton further in front before Kevin Long set up a tense finale with his first goal for Burnley late on.
The Championship high-flyers were unable to force a replay, though, as their top-flight opponents won for just the second time in 10 games in all competitions.
Southampton made five changes from the New Year's Day defeat to Chelsea, Lambert among those to return at the expense of Rodriguez, while the visitors made two alterations in defence as Daniel Lafferty and Long came in.
Burnley threatened as early as the first minute, Scott Arfield sending in a curling effort from the corner of the box that flew just wide of the post.
Clyne gave his side the lead midway through the half in spectacular fashion, thumping a right-footed effort into the top corner from 25 yards for his first goal of the season.
Lambert doubled Southampton's advantage six minutes later as he picked the ball up in space on the edge of the area and unleashed another unstoppable drive beyond Tom Heaton into the top right-hand corner.
The England international's afternoon turned sour 10 minutes before the break, however, when he hobbled off with a knock to be replaced by Rodriguez.
Heaton dived to his right to keep out Luke Shaw's effort in the 41st minute as Southampton finished the first half strongly.
Six minutes into the second half, the visitors pulled one back as Southampton-born Vokes flicked home Kieran Trippier's right-wing cross for his 11th goal of the season.
Burnley drew level six minutes later as Ings cut in from the right and squeezed the ball between Kelvin Davis and the near post from a tight angle to take his tally for the season to 20.
Vokes almost completed a remarkable turnaround immediately following the equaliser, but his effort was superbly blocked by Davis.
And Vokes was made to pay for that miss in the 66th minute, Rodriguez sweeping home after the ball had bounced around the Burnley penalty area.
As the game became increasingly open, Ings came close to a second equaliser in the 72nd minute when his low drive came back off the post and Lallana went down the opposite end, advanced into space and fired low beyond Heaton from 20 yards.
Long turned in Junior Stanislas' corner three minutes from time as Burnley continued to battle right to the end, but Southampton hung on to progress.