The forward joined fellow Three Lions World Cup hopefuls Adam Lallana and Rickie Lambert on the scoresheet as Mauricio Pochettino's men cruised to victory
Lambert unselfishly squared for Rodriguez to score the first before getting on the scoresheet himself early on in the second half.
|VIEW FROM ST MARY'S
|By George Ankers
"That's an extremely convincing win for Southampton as they take what should be a decisive hold on eighth place and retain their hopes of aiming higher. Newcastle started competently enough but were beaten down after an onslaught of first-half chances - they look like they've had enough of this season. The hosts' English contingent were all excellent, with Adam Lallana the star for his creativity and thumping third goal, while Jay Rodriguez and Rickie Lambert both pressing their World Cup claims in serious fashion. The home supporters are delighted and they have every right - they've witnessed some really, really good football."
Reserve Newcastle goalkeeper Rob Elliot, in for the injured Tim Krul, produced a string of outstanding saves in the first half, but his efforts proved in vain in a dismal showing from the rest of the Newcastle side.
Alan Pardew started his four-game touchline ban following the end of his three-match stadium suspension for headbutting Hull City's David Meyler earlier this month but the visitors started poorly without his presence in the dugout.
Elliot saw his goal threatened three times in the early going by Rodriguez, who missed a glaring opportunity after breaking free of the Newcastle defence, before then heading wide from a Calum Chambers cross.
Rodriguez did eventually produce an effort on target with a low header, but was thwarted by a fine diving save from Elliot.
That stop clearly boosted Elliot's confidence and he again came to the rescue after 15 minutes with an excellent one-on-one save from Lambert.
Southampton continued to waste opportunities, with captain Lallana somehow heading wide from point-blank range after Chambers had knocked a clever ball from Lambert into his path.
The hosts' dominance was evident in every facet of the game, with only Elliot, who produced a marvellous stop shortly before the break to again deny Rodriguez, keeping the scores level.
However, Elliot was finally beaten in first-half injury time as Rodriguez slotted home a simple finish after Lambert had sprung the visitors' offside trap.
Newcastle made two changes during the interval, Davide Santon and Luuk de Jong replacing Mapou Yanga-Mbiwa and Hatem Ben Arfa.
But those alterations had little impact and Southampton doubled their lead just four minutes into the second half when Lambert rattled home Lallana's right-wing cross at the far post.
The visitors were lucky to keep 11 men on the field following a rash challenge on Jack Cork from Papiss Cisse, but their misery was soon compounded by Lallana and Rodriguez.
First, Lallana picked up the ball in midfield, surging forward before unleashing a vicious long-range effort that flew past Elliot into the top corner, before Rodriguez emphatically converted Gaston Ramirez's ball over the top of the Newcastle defence a minute from time.