The 23-year-old joined from Burnley in the summer for what was a club record fee and today showed why Saints forked out £6million for his services.
Rodriguez 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.
Liverpool fan Rickie Lambert soon doubled Saints' advantage with his 13th goal of the season coming when his 25-yard free-kick deflected home off Daniel Sturridge.
However, as good as Southampton are on the attack, they have often been found wanting at the back and 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.
Liverpool's regular number one Pepe Reina was unavailable for Saturday's match, meaning understudy Brad Jones once again deputised in goal.
The Australian conceded twice in last weekend's defeat of Tottenham and had to pick the ball out of his goal again in the sixth minute.
After earlier attempts by Rodriguez and 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 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.
The Australian was again called into action to deny a stinging Rodriguez drive moments after dealing comfortably with a Ramirez effort.
Liverpool were struggling to cope with Southampton's attacking intent and failing to create anything of note at the other end.
Aside from an offside Sturridge effort two minutes in, Rodgers' side had little to show for their efforts as Saints' defence dominated.
Liverpool's backline were enduring a much tougher afternoon, with Rodriguez hooking over from close range after Jones parried an Adam Lallana strike.
The Reds were struggling to cope with the home side's wing play but beginning to make inroads at the other end.
Luis Suarez was incensed by referee Phil Dowd's decision to call back play for a foul on Jos Hooiveld when he was through on goal, before Sturridge struck from distance.
Liverpool will have been relieved to see the impressive Hooiveld limp off soon after, 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 of Sturridge.
A lunge from Fonte lunge denied Daniel Agger an immediate response, before an exquisite Rodriguez snapshot flew inches wide at the other end.
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 Suarez almost latched onto a through ball by substitute Lucas.
It seemed to jolt Saints back into life as they began threatening down the flanks again, although Suarez was still proving a thorn in the side at the back.
Boruc saved a free-kick from the Uruguayan but Saints were well and truly on the back foot, with only latch-ditch defending and fine goalkeeping denying the visitors.
Boruc stopped a long-range Gerrard effort, before Jones did fantastically to smother a goalbound Lallana dink.
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 Skrtel and tucked home after Jones stopped his initial effort.
Jose Enrique put into the side-netting as Liverpool looked for a late goal, but the only late action was a booking for Suarez after hassling Boruc.