Morgan Schneiderlin and Rickie Lambert fired Saints into a two-goal lead but Philippe Coutinho, a reported January target for the hosts, reduced the deficit in first-half stoppage time.
Brendan Rodgers' side pushed hard for a leveller but Rodriguez made sure of the three points with 10 minutes remaining, putting Saints seven points clear of the drop zone in front of a record Premier League crowd at St Mary's of 32,070.
After earlier attempts by Rodriguez and Gaston Ramirez, the pair combined brilliantly to set-up the opening goal in the sixth minute. 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 and they could - and probably should - have doubled their lead in the 10th minute. Ramirez put Saints talisman Lambert through on goal but he could not find a way past Jones.
Aside from an offside Daniel Sturridge effort two minutes in, Rodgers' side had little to show for their efforts as Saints' defence dominated.
Liverpool will have been relieved to see the impressive Jos Hooiveld limp off, with Jose Fonte coming on in his place after 30 minutes.
However, their task was made all the more difficult three minutes later when Southampton netted a fortuitous second. After Martin Skrtel was adjudged to have fouled Lambert, the Saints striker stepped up to hit the 25-yard free-kick that wrong footed goalkeeper Jones after taking a deflection off 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.
Southampton were, in truth, feeding off scraps in the second half but got the goal they desired, thanks to fine work by Rodriguez. The former Burnley striker burst through own goal, brilliantly drove past Skrtel and tucked home after Jones stopped his initial effort.
Source: PA
Source: PA