Rodriguez stepped off the bench to score the winner with two minutes remaining thanks to a cool finish.
The 19-year-old rounded the goalkeeper following a superb through ball from Chris Eagles, who had also stepped off the bench.
It means Owen Coyle's men move into the fourth round draw after stretching their unbeaten run to six games.
However the Clarets found themselves on the back foot in only the fourth minute when Andy Johnson skipped away, only for his cross to be diverted to safety by Clarke Carlisle.
Johnson then tried to pick out Clint Dempsey but Clarke made another vital interception for the home side in the penalty area.
Dempsey was seeing plenty of the ball and tried to chip Brian Jensen in the 17th minute, only for the goalkeeper to make a comfortable save.
Fulham were looking dangerous on the break and three minutes later Stephen Jordan did well to block Johnson's teasing delivery from the left.
Burnley responded and Martin Paterson made space in the area but sent his effort over the bar.
The home side almost took the lead in the 31st minute when Chris McCann latched on to pass from Paterson - only for goalkeeper Pascal Zuberbuhler to get a vital touch.
Three minutes later the goalkeeper was put under pressure by team-mate Chris Baird's sloppy backpass.
The pacey Paterson tried to take advantage but Zuberbuhler made a crucial block to spare Baird's blushes.
Fulham hit back and Paul Konchesky came close in the 43rd minute following a quick free-kick from Johnson.
The full-back rattled in a shot from 20 yards but Steven Caldwell made the clearance.
Fulham started the second half brightly and Dempsey's effort went narrowly wide in the 56th minute after he had been set up by Gera.
The visitors kept up the pressure and a cross from Konchesky was cut out by the impressive Carlisle.
Burnley were penned back again on the hour when Zoltan Gera made room for himself, only to see his effort blocked.
The Clarets responded in the 62nd minute with a swift counter-attack and McCann rattled in a shot which was cleared by Gera.
From the rebound McCann saw his effort go past the goalkeeper but past the post as well.
Fulham launched a swift-counter attack in the 68th minute but Seol Ki-Hyeon failed to get a good enough touch to trouble Jensen.
McCann squandered a great chance of giving Burnley the lead in the 77th minute in front of a crowd of 7,119.
He peeled away from his marker following a cross from Stephen Jordan but he sent a soft header into the goalkeeper's arms.
Rodriguez, however, had the final word as extra-time loomed to leave Fulham stunned.
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