Nathaniel Clyne and Rickie Lambert put the Premier League hosts in complete control at St Mary's, but Sam Vokes and Danny Ings hit back for the Clarets in an entertaining encounter which also saw Adam Lallana and Kevin Long get on the scoresheet.
Burnley could do little but stand and admire the opening goal of the contest, which arrived on 22 minutes, with Clyne crashing a spectacular 25-yard half-volley high into the top corner.
Clarets goalkeeper Tom Heaton was reduced to little more than a spectator role again six minutes later as England international Lambert gathered on the edge of the box and drilled into the back of the net.
Burnley are flying high in the Championship, though, and boast two in-form frontmen in Vokes and Ings.
The former handed them a route back into the tie when he nodded home a Kieran Trippier cross on 51 minutes and six minutes later Ings got in on the act as he fired past Kelvin Davis from a tight angle.
The tide was, however, to turn back in Southampton's favour, with Rodriguez turning in his 10th goal of the season following a scramble in the Burnley penalty area just past the hour mark and Lallana adding a fourth on 73 minutes to ensure that a late effort from Long counted for little in the grand scheme of things.
Burnley proved a threat from the outset, with Scott Arfield swinging a strike just wide after 43 seconds.
Vokes and Ings also had early chances, although it was Mauricio Pochettino's men that were enjoying the lion's share of possession.
Gaston Ramirez was involved in much of the early play and sent in a fine cross, which looked set to be nodded home by James Ward-Prowse until Danny Lafferty's intervention.
The Clarets were showing just why their defensive record is the second best in the Championship, although they could do little to prevent the opening goal.
Clyne was the unlikely source, collecting a pass before sending a rasping half-volley into the top right-hand corner from the edge of the box.
Six minutes after Clyne netted, Saints talisman Lambert impressively struck a second into the top-right hand corner after picking up the ball 20 yards out.
Lambert wasted a chance to make it 3-0 soon after - his last action before limping off.
There were few more popular figures inside St Mary's than Lambert's replacement Rodriguez, whose 35th-minute introduction was greeted by a loud ovation by all.
There was little more action before half-time, from which Burnley returned strongly.
Six minutes after the restart Trippier swung in a cross from the right and Southampton-born Vokes impressively directed his header past Davis.
Heaton easily dealt with a Rodriguez strike soon after, before Burnley drew level through another Hampshire native as Ings showed impressive skill, control and confidence to cut inside and lash home from an acute angle.
Lallana, splashed across the back pages on Saturday due to the Mark Clattenburg case, was brought on in a bid to change Saints' fortunes, with Rodriguez seeing a strike saved by Heaton when they returned to the attack.
The Burnley-born forward would not be kept off the scoresheet for long, though, slotting into the empty net as Ward-Prowse took a tumble in the box.
Play was swinging from end to end and Ings rattled the post as Burnley threatened an equaliser, although Lallana put the game out of sight with 17 minutes left as he drove forward to unleash a fine, low strike from distance.
Despite the goal, the visitors clearly still believed they could force a replay, with Ings threatening before Long bundled home their third of the afternoon.
Buoyed by 1,845 travelling fans, Burnley continued to press but there was to be a no grandstand finish as the Saints secured a first win of 2014.