The Clarets faced the threat of a 15th game without a win at home in all competitions when their League Two opponents took a surprise lead.
Jake Beesley tucked the ball into the corner just two minutes into the second half as the visitors looked to spring a cup upset.
However, Burnley rectified their first half profligacy with a clinical display in front of goal thereafter.
Maxwel Cornet ticked off his first assist for the club when Rodriguez converted his corner with a thumping header.
The Ivorian, who had chance after chance to open his account for the Clarets, was then instrumental as the hosts got their noses in front.
The ex-Lyon winger's cross was cushioned into the path of Rodriguez by Matej Vydra and the striker tucked the ball into the corner.
The 32-year-old brought his hat-trick up in the 62nd minute when his glancing header from Ashley Westwood's delivery beat Jay Lynch.
His fourth and final goal was the easiest of the bunch as he turned the ball home from close range following good work from Aaron Lennon.
Here are the ratings.
9. Jack Cork 6.5
Enjoyed the hustle and bustle of a midfield battle with Stephen Dooley and Aaron Morley, limiting their time on the ball. Pressed hard, outworked his opponents, broke up the play and carried the ball well. His efforts afforded Ashley Westwood more time to get on the ball.
10. Jay Rodriguez 8.5
Just over a decade since his four goals against Burton Albion in this competition, the striker was at it again. Took all of his goals really well, with two exquisite headers, and surely staked a claim for a start away at Leicester. Four-midable performance!
11. Matej Vydra 7
More than played his part in a comfortable win at Turf Moor. Chased everything, stretched the play, almost teed up Jay Rodriguez in the first half having thumped an effort wide of the near post beforehand. Needs more composure in possession.