The Dutch winger, on loan from Werder Bremen, scored his first goals for Ronald Koeman's side either side of a bizarre strike for Yoan Gouffran at St James's Park
The Dutch winger slid the first past an unimpressive Tim Krul just before the 15-minute mark, having been played in by James Ward-Prowse, but the Magpies had an equaliser in bizarre fashion another quarter-hour later when Florin Gardos's rash clearance bounced off Yoan Gouffran's shin and into the net.
Elia snagged his second after the break when, pitted against Daryl Janmaat by Graziano Pelle's canny headed flick, he found enough room to shoot, with a heavy deflection from his countryman diverting it into the goal.
|VIEW FROM ST JAMES'S PARK
|By Michael Lightfoot
The Saints were made to work hard for their victory but they nevertheless go marching back into third place in the table. Newcastle were better for most of the game but they left their finishing boots in the dressing room.
It was not the best performance from Southampton but, if they can play badly and still win, then that just shows how good a team Ronald Koeman has put together. As for the Magpies, they drop down into 11th and are in serious need of a new manager, despite giving the Saints a game here.
Elia darted between the returning Massadio Haidara and makeshift centre-back Paul Dummett to collect Ward-Prowse's pass and squeeze a shot through Krul, possibly caught out by Janmaat's sprawling challenge.
Ward-Prowse was the provider again in the 24th minute, this time with a perfectly flighted free kick as Jose Fonte headed over before Newcastle levelled in fortuitous fashion. Gardos slid to cut out Remy Cabella's deflected through-ball, his interception cannoning into Gouffran and looping into the net.
The goal put Newcastle back in the ascendancy for the remainder of the half and a stretching Ayoze Perez almost put them ahead on the end of Cabella's lofted 43rd-minute pass.
Full-back Haidara whipped in a wonderful cross for Gouffran to head wastefully wide in the 61st minute - a miss which Newcastle would soon regret.
Shane Long, on for ineffective Old Trafford hero Dusan Tadic, won a flick-on which Elia was alert to collect and the winger's shot deflected of Janmaat to leave Krul with no chance.
Dummett wasted a free header from Jack Colback's 75th-minute corner as it became clear that Carver's quest for a maiden win in the St James's Park hotseat was set to continue.
Southampton striker Graziano Pelle glanced a cross from the similarly impressive Nathaniel Clyne against the crossbar with 12 minutes remaining before Koeman's men closed their fifth win in six unbeaten league games.
Newcastle made them sweat in stoppage time, with Cabella missing his kick as Sammy Ameobi cut the ball back into the box, while fellow substitute Emmanuel Riviere clattered a shot into Southampton captain Fonte's outstretched arm but referee Robert Madley rejected desperate appeals for a penalty before whistling for full-time.