Austin recently joined Southampton from Championship side Queens Park Rangers and made his debut against Manchester United away from home on Saturday afternoon.
The 26-year-old striker came on as a substitute in the 79th minute and scored the only goal of the game on 87 minutes.
Over the course of the 90 minutes at Old Trafford, hosts United had 56% of the possession, took eight shots of which one was on target, and earned four corners.
Visitors Southampton had 44% of the possession, took seven shots of which two were on target, and earned four corners.
Ronald Koeman’s side are eighth in the Premier League table at the moment with 33 points from 23 matches.
The Saints are nine points behind Tottenham Hotspur, who are currently fourth and occupy the final Champions League spot.