The club record signing put Mauricio Pochettino on course for a second win at the helm in the sixth minute, nodding a Gaston Ramirez cross into Morgan Schneiderlin's path to slot home.
Rickie Lambert soon doubled Saints' advantage and although Philippe Coutinho, a reported January target for the hosts, reduced the deficit in first-half stoppage time, Rodriguez made sure of the three points with 10 minutes remaining.
The hosts started brightly and after earlier attempts by Rodriguez and Gaston Ramirez, the pair combined brilliantly to set-up the opening goal.
Ramirez floated a weighted cross to the back post to Rodriguez, who nodded across the face of goal for Schneiderlin to direct past Brad Jones.
It was quite a start by Saints, who netted a fortuitous second when Lambert stepped up to hit a 25-yard free-kick that wrong footed goalkeeper Jones after taking a deflection of Daniel Sturridge.
It was Liverpool that ended the half on top, though. After Artur Boruc denied Coutinho on a one-on-one, the Brazilian beat the Saints goalkeeper.
Steven Gerrard headed across goal to Sturridge, whose blocked shot was slotted home by the former Inter Milan man.
It was the 18th goal Saints have conceded in the final 15 minutes of the first half this season and Coutinho found the back of the net again five after the restart, although he had already been ruled offside.
Sturridge was next to come close, hitting a half-volley into the side-netting before Luis Suarez almost latched onto a through ball by substitute Lucas.
Southampton were, in truth, feeding off scraps but soon got the goal they desired, thanks to fine work by Rodriguez. The former Burnley striker burst through own goal, brilliantly drove past Martin Skrtel and tucked home after Jones stopped his initial effort.