The Dutchman, on loan from Werder Bremen, scored his first goal for the club with 14 minutes gone and then after Yoan Gouffran had cancelled out his strike with a fortuitous 29th-minute equaliser, restored the visitors' lead 17 minutes into the second half.
Both goals came against the run of play as Newcastle, who have now played four games under caretaker manager John Carver and won none of them, dominated possession for long periods
However, they lacked the cutting edge.
However, the hosts were aggrieved not to be awarded an injury-time penalty when substitute Emmanuel Riviere's shot appeared to hit Jose Fonte's arm - but referee Robert Madley waved away their appeals..
That will matter little to the Saints, who maintained their push for an unlikely top-four finish in front of a crowd of 49,307 at St James' Park.
By contrast, Carver and his players will head off on a mid-season training break in Dubai with the search for a new head coach ongoing and their season threatening to unravel.
Vurnon Anita sliced a second-minute shot wide after good build-up play by Moussa Sissoko and Remy Cabella, before full-back Daryl Janmaat saw his well-struck effort blocked after playing a neat one-two with Cabella
Ayoze Perez and Gouffran then tested goalkeeper Fraser Forster from distance, before Southampton opened the scoring.
The impressive Graziano Pelle headed down Fonte's long ball for James Ward-Prowse to slide in Elia, whose well-struck 14th-minute shot embarrassed keeper Tim Krul.
With Pelle proving a handful for central defenders Fabricio Coloccini and Paul Dummett, the visitors gained a measure of control as the home side attempted to regroup, and Fonte headed Ward-Prowse's 25th-minute free-kick over.
But Newcastle were level within four minutes with the help of a large slice of good fortune.
Defender Florin Gardos looked to have Cabella's through-ball covered, but his attempted clearance cannoned off the chasing Gouffran's shin and looped past Forster and into the unguarded net, the Frenchman's first goal since Boxing Day 2013.
Forster was relieved not to be beaten for a second time three minutes before the break when he could only look on as Perez's cushioned volley from Cabella's swinging cross dropped just wide.
Carver tweaked his team at the break with Sissoko moved into a central position for the second half, and after an opening period during which Southampton dominated, the Magpies gradually rediscovered their rhythm.
Cabella sliced over from Sissoko's 53rd-minute knock-down, but the Saints responded with Pelle once again the focal point.
Ronald Koeman sent on Jack Cork and Shane Long for Ward-Prowse and Dusan Tadic on the hour, but his side might have been behind within seconds when Gouffran met Massadio Haidara's cross with a glancing header, but steered his effort wide of the far post.
However, Southampton regained their lead within seconds when Long flicked on a clearance and Elia slipped through the heart of the home defence to fire home off Janmaat's shins.
Carver responded by sending on Riviere and Sammy Ameobi for Anita and Gouffran, and the traffic was largely one-way with Dummett passing up a glorious opportunity to level 15 minutes from time when he met Jack Colback's corner unopposed, but headed harmlessly wide.
Pelle was significantly more accurate two minutes later when he powered a header from full-back Nathaniel Clyne's cross against Krul's bar, but as time ran down, the home side committed men to the search for a second equaliser which might have come had Madley ruled that Fonte had handled Riviere's effort.
But ultimately, their efforts proved to be in vain as the Saints held out with little difficulty to extend their recent run to 16 points from a possible 18.
Source : PA
Source: PA