The Saints started the match brightly and carved out the first chance of the afternoon in the 11th minute when Dusan Tadic volleyed Sadio Mane's cross just wide of the post.
Martial inspires Manchester United to victory at Southampton
Anthony Martial scored twice on his first Premier League start as Manchester United emerged 3-2 winners at Southampton.
The early warning went unheeded and United found themselves trailing minutes later, Graziano Pelle scrambling home the rebound after Mane's initial strike was parried by David de Gea.
United's defence was caught ball watching again in the 16th minute, allowing Pelle to race clear for a one-on-one with De Gea, but the Italian's effort came back off the post.
Southampton dominated the opening stages, but United had the chance to level thing up in the 25th minute when Juan Mata's lofted pass found the head of Memphis Depay in front of goal, only for the Dutchman to head over unchallenged.
The Red Devils were on level terms in the 33rd minute courtesy of Martial on his full league debut. The French youngster latched onto a through-ball from Mata and wrong-footed Maarten Stekelenburg by feigning to shoot, before finishing on the turn.
Ronald Koeman's side looked rattled after surrendering the lead, and United pressed for a second as Mata went on a mazy run into the danger zone, but a well-timed challenge from Virgil van Dijk stopped the final attack of the first period in its tracks.
United took the lead for the first time in the match just after half time when an under-hit back-pass from Maya Yoshida sent Martial clean through on goal, and he kept his composure to double his tally for the afternoon.
Southampton almost drew level on the hour mark as Jose Fonte leaped high in the box to send a bullet header towards goal, which De Gea only just kept out at full stretch.
Louis van Gaal's side extended their lead in the 68th minute through Mata, who was on hand to rifle home the rebound after Depay crashed an effort against the upright.
The closing stages were anything but comfortable for the Red Devils as Pelle notched up his second of the match with five minutes to go, the Italian planting a Mane cross beyond De Gea to set up a nervy finish.
United had five long minutes of stoppage time to hold out through, and Southampton almost made the count as Victor Wanyama tested De Gea with 93 on the clock, but the Red Devils made it over the line to collect maximum points.
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