PLAYER RATINGS: Burnley 4 Rochdale 1
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.
1. Nick Pope 6
Called into the side after Wayne Hennessey sustained an injury in training. Handling was impeccable, kept hold of everything that came into the penalty area, and made a good block early on to keep out Jake Beesley.
2. Phil Bardsley 5.5
A rather erratic display from the Burnley full back. Slightly off the pace and given plenty to do by Danny Cashman and the marauding James Keohane in the first half. His workload eased after the break and he was able to get forward in support of Aaron Lennon.
3. Erik Pieters 5.5
Given the run-around by pacey forward Abraham Odoh in the first half. Dallied on the ball at the back, occasionally putting his side under unnecessary pressure, but got forward well. Vast improvement in the second half and distribution went up a notch.
4. James Tarkowski 5.5
The defender, who inherited the captain's armband for the night, wasn't at his best, but didn't need to be. Far too casual at times, conceded a cheap free kick in a dangerous position when tripping Jake Beesley, and poor pass out from the back led to the goal.