Jay Beesley had smartly given the visitors the lead before Rodriguez equalised minutes later.
And Jay-Rod got his second shortly followed by his third around the hour mark to turn the tie on it’s head.
And he added another to repeat a feat he achieved against Burton Albion in the same competition ten years ago, getting his fourth before being withdrawn.
Story of the game
Burnley responded with a chance of their own through Nathan Collins watched his header go wide of the post before fellow summer arrival Maxwel Cornet got in behind but couldn’t lift his effort over Dale’s goalkeeper.
Ashley Westwood flashed his shot past the post from outside the area for the hosts before Aaron Morley couldn’t keep his free kick down as the League Two side pushed for a surprise opener.
The Clarets had their best chance of the half soon after when Jay Rodriguez and Vydra played some neat football and Cornet trickled his effort wide, but he really should have hit the target.
Jack Cork then saw a tame curling effort well held by Lynch in a first half that saw plenty of chances but poor finishing.
Jimmy Keohane curled an effort over at the other end just before the half time whistle.
Rochdale took an age to re-emerge after half time and their slight delay maybe caught Burnley out, as Beesley opened the scoring.
The striker took it around once before hitting it along the ground across Pope and into the corner.
Their lead didn’t last for long as Burnley hit back through Rodriguez.
Cornet’s corner was met at the near post by the Clarets’ man who flicked a header towards goal over Lynch and into the net.
Vydra saw his header saved well moments later as the Premier League side pushed for a second.
Rodriguez did get that second with an effort from inside the box to complete the hosts turnaround.
And the former Clarets youth player got his third of the game – completing a 12-minute hat-trick – with another header from a corner.
And the game was over as Rodriguez grabbed his fourth 13 minutes from time, finding the net from close range.
Cornet kept going and set up Rodriguez’s replacement Chris Wood for a good opportunity but Lynch made another save.
The game slowly fizzled out and Burnley progressed to the fourth round of the cup, but it could have been more.
Takeaways from the game
There’s not to much you can take from tonight’s game on the balance of things: Burnley should always be beating Rochdale comfortably.
However, Maxwel Cornet is one who certainly impressed throughout the game with the sheer volume of chances he had, especially in the first half where, on a different day, he could have scored at least three himself.
Obviously, the night belongs to Rodriguez.
He took his chances with great confidence and his movement in and around the area caused a number of problems.
Himself and Vydra were an issue throughout, especially down the left side with Cornet.
Conceding won’t be a major concern for Sean Dyche but he will have food for thought with how much of a fight Rochdale put them through.
The visitors can be proud of their performance, showing a lot of grit to keep themselves level and Beesley’s finish was a calm, assured one.
But, it was Burnley who progress thanks to their first home win since January and they will be in tomorrow’s draw.