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.
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