The visitors took the lead against the run of play in the 14th minute of the Premier League clash when Elia shot past Tim Krul for his first Saints goal, before Newcastle levelled in bizarre circumstances when Florin Gardos’ clearance hit Yoan Gouffran and found the net.
Elia restored Southampton’s lead when the loanee’s deflected shot found the far corner, before Graziano Pelle almost added a third when his header rattled the crossbar.
Newcastle piled on the pressure late on as they searched for an equaliser, but Ronald Koeman’s side held on to secure the points.
The hosts started brightly with Ayoze Perez striking a low effort straight at Fraser Forster, before the Saints goalkeeper had to be alert again soon after to deal with Gouffran’s dipping half-volley.
Southampton opened the scoring when Pelle controlled with his chest to find James Ward-Prowse, who released a weighted pass for Elia to fire home under Krul from inside the area.
The visitors came close to doubling their advantage a few minutes later, but Jose Fonte headed over from close range after beating Fabricio Coloccini in the air from Ward-Prowse’s free-kick.
Newcastle levelled in the 29th minute, when a lack of communication between Gardos and Forster when dealing with Remy Cabella’s through ball saw the defender’s clearance ricochet of an unsuspecting Gouffran inside the area and loop in to an empty net.
The home side started the second half positively and almost took the lead just after the hour mark, but Gouffran headed Massadio Haidara’s cross wide.
The visitors restored their lead soon after, as Shane Long beat Coloccini in the air to find Elia, whose effort deflected off Daryl Janmaat’s leg and found the far corner.
John Carver’s side hit back with a series of corners in quick succession, with Paul Dummett heading wide unmarked from six yards out.
Pelle came close to killing the game off the Saints in the 77th minute, but headed Nathaniel Clyne’s pin-point cross against the woodwork.
Subs: Williamson (6), Ameobi (6), Riviere (6)
Southampton: Forster (6), Clyne (7), Gardos (6), Fonte (6), S. Davis (7), Tadic (6), Ward-Prowse (8), Pelle (7), Bertrand (6), Elia (9), Reed (6).
Subs: Long (6), Cork (6), Targett (6)
Man of the Match: Eljero Elia