Chelsea loanee Nathan Ake, who recently scored a last-gasp winner against Liverpool, headed the home side in front in the sixth minute from an excellent cross by Jack Wilshere.
Southampton, who had made six changes amid their busy schedule, equalised eight minutes later with a low drive by England fullback Ryan Bertrand.
An inspired halftime change brought on Steven Davis to set up Rodriguez, who then made sure of the win five minutes from the end with his second goal, a superb effort from distance.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
The result sent Southampton up two places to seventh, three points ahead of Bournemouth, who stayed 10th.
It was only the 19th league meeting between the clubs, situated some 50 kilometres apart, with Southampton having almost invariably been in a higher division.
Against opponents, who have never been higher in their history, they were on the backfoot early on, however, and soon fell behind.
The latest in a series of Bournemouth corners was cleverly worked short on the right-hand side before Wilshere crossed to the far post where Ake dived in front of Pierre Hojbjerg to head in.
It was poor defending by the Dane, one of the six players manager Claude Puel had brought into the side for his team's eighth match in less than a month.
Yet Southampton were rarely in trouble for the rest of the game. They equalised within eight minutes when Bertrand was allowed to cut in from the left flank and drive a low shot from a difficult angle across goalkeeper Artur Boruc.
It was the former Chelsea defender's first goal of the season and meant Saints were no longer the joint lowest-scoring side in the league.
Their defensive record remains one of the best and Bournemouth struggled make any threatening opportunities.
Northern Ireland international midfielder Davis made an immediate impact after replacing Harrison Reed, who had made a rare start.
Barely two minutes into the second half, Davis played a neat one-two and set up Rodriguez to slide in his first league goal since August.
Despite being without leading scorer Charlie Austin, who is expected to be out for three or four months, the visitors were close to scoring again through Rodriguez, Nathan Redmond and Hojbjerg, whose shot was cleared off the line by Ake.
Rodriguez's second goal was an outstanding finish as he swivelled to beat Boruc with a fierce shot from 20 metres.
That rounded off an excellent second-half performance by his side that left them worthy winners.
(Reporting by Steve Tongue; Editing by Toby Davis)
