The home side rushed out of the blocks and Mathieu Debuchy saw his fourth-minute volley fly wide of the post.
Only a goal-line clearance from Calum Chambers prevented Loic Remy from giving the Magpies an 11th-minute advantage, after Gouffran had nicked the ball from the grasp of Paulo Gazzaniga to set up his compatriot.
Southampton struggled to create any goalscoring chances and were restricted to a long-range effort from James Ward-Prowse.
© Getty Images
Newcastle took the lead in the 27th minute when Jos Hooiveld swiped at thin air when attempting a clearance, which left Gouffran with the task of rounding a stranded Gazzaniga and tapping the ball into an empty net.
The hosts continued to pen the visitors inside their own half, and Vurnon Anita beat two defenders before flashing a good ball across the face of goal, but no Newcastle attackers were on hand to tap it home.