The match seemed certain to end in stalemate until Rodriguez headed Ashley Westwood’s 89th-minute cross home, the only on-target effort by either side in the match.
A Cherries side that was unchanged from the one that beat Chelsea at Stamford Bridge last weekend almost lost Simon Francis in the opening minutes when he was caught by Ashley Barnes’ boot, although he was eventually able to continue with a bandage on his head.
There were very few chances in a drab first period, with Jefferson Lerman’s 30-yard drive, which flew wide, being a rare highlight.
James Tarkowski and Ryan Fraser were involved in an aerial challenge which left the Cherries star with a bloody nose before Aaron Ramsdale had to cope with some dangerous crosses as the first half drifted to a close.
Eddie Howe brought on Callum Wilson for the second half, which was played in heavy rain, and Bournemouth stepped up the pace, with Clarets captain Ben Mee doing well to stop a Jack Stacey cross.
Nick Pope dealt with a dangerous Fraser cross but a rash of yellow cards midway through the second half made sure that there was no flow to the encounter.
Dan Gosling fired a weak effort wide, Chris Mepham failed to find the target with a header and Lerma sent a volley wide, but it was the Clarets who picked up the three points thanks to substitute Rodriguez’s late header.