West Ham arguably dominated the early stages of the contest, but it was the Saints who took the lead after 10 minutes, as Manolo Gabbiadini combined for a one-two with Nathan Redmond in order to slot beyond Joe Hart and end the club's six-match goal-scoring drought.
Marko Arnautovic made West Ham's task of earning points on the road even more difficult 20 minutes later, as the club's summer signing saw a straight red for a needless elbow on a helpless Jack Stephens.
The Saints soon went on to capitalise on proceedings not too long after, this time as Jose Fonte dragged down Steven Davis inside the area, allowing Dusan Tadic to slot home down the middle despite the efforts of Hart.
The Hammers did, however, give themselves a lifeline just before the break, when Javier Hernandez reacted quickest to Michail Antonio's parried shot to bundle home and leave the score at 2-1 going into half time.
Gabbidini might have earned himself a brace and handed his side some breathing space, but the forward could only force his effort against the crossbar with Hart looking beaten in the West Ham goal.
The Saints were soon made to pay for their profligacy, as Hernandez stunned the home fans after prodding home a second of the afternoon for the visitors, having made use of Diafra Sakho's header which had originally cannoned back off the bar.
Southampton were somehow able to still muster all three points from the contest, though, as Austin fired home from the spot in order to hand Mauricio Pellegrino his first league victory while in charge on the south coast.