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.
However, Burnley rectified their first half profligacy with a clinical display in front of goal thereafter.
5. Nathan Collins 5.5
Caught out by the pace of Abraham Odoh and the physicality of Jake Beesley, who left the defender for dead inside the box when giving Rochdale the lead. However, picked up some good positions to clear crosses when Dale's forwards got beyond the full backs.
6. Aaron Lennon 7.5
A lively and energetic display from the ex-England international, who showed Jeriell Dorsett a clean set of heels time and time again. The winger, who saw a ferocious volley blocked, was instrumental in Burnley's fourth and final goal.
7. Maxwel Cornet 7.5
Squandered countless opportunities to get off the mark for the Clarets having been played through on goal several times. Claimed his first assist, however, and had a big hand in Burnley's second goal of the evening. Ivorian also diligent in his defensive duties.
8. Ashley Westwood 6.5
Switched the play remarkably well, allowing his team-mates to get on the ball in space. Mixed his play up in the middle of the park with an effective array of passing and set up Jay Rodriguez for his hat-trick before being withdrawn late on.