The 26-year old Austin, who joined the Saints from Queens Park Rangers last week, hammered the ball past keeper David De Gea after an inviting cross.
Southampton repeated their success in the corresponding fixture last season when a Dusan Tadic goal gave them their first league victory at Old Trafford in 27 years.
Following a smash-and-grab 1-0 win at fierce rivals Liverpool last week, United lacked spark and creativity.
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Their afternoon was summed up by several harmless efforts by lone striker Wayne Rooney.
(Writing by Zoran Milosavljevic, editing by Ed Osmond)
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