Striker Austin was in the stands at St Mary's, after completing his Â£4million move from QPR, to watch James Ward-Prowse triple his Premier League goal tally in a 3-0 win over West Brom.
England Under-21 midfielder Ward-Prowse struck with a free-kick and a penalty to put Saints two up at the break before Dusan Tadic added the third in the second half.
Boss Koeman, who hopes Austin will provide "goals and competition" for the club, is hoping the floodgates have now opened for Ward-Prowse.
"The free-kicks are one of his qualities," he said
"We tell him to keep working on them in training because he can score out of that situation.
"It's important for the boy, it will lift his confidence
And that free-kick was so important to break open the game
Then when you are 1-0 up West Brom have to do more and do things differently."
Just five minutes had passed when Craig Dawson fouled Sadio Mane and Ward-Prowse curled his free-kick inside Boaz Myhill's near post.
Dawson was again the guilty party in the 33rd minute, although the defender was perhaps unlucky that his collision with Matt Targett in the area was adjudged a foul.
Nevertheless, with regular penalty taker Tadic on the bench, Ward-Prowse did the honours with a coolly taken spot-kick which sent Myhill the wrong way.
Tadic entered the fray just after the hour and took only eight minutes to get his name on the score sheet and wrap up an emphatic victory.
"It was a comfortable win," added Koeman
"We know how difficult is to beat West Brom
It's a very good result.
"We had total control of the game, they didn't create many chances and we had chances to score more
It was a very good team performance."
Albion manager Tony Pulis blamed his side's lethargic performance on having to play back-to-back away games following the midweek draw at Chelsea
"Southampton were the better team," he admitted
"We looked flat, we had no spark,
"But I'd be interested to see how many teams have played their second game away from home this weekend and picked up any points.
"The lads looked tired, and we've done well away from home this season
But just after Christmas, two away games is very difficult
Why not have one away and then one at home?
"But take nothing away from Southampton
It was a bad day at the office for us, but we'll dust ourselves down and move on."
Albion skipper Darren Fletcher went off with a knee injury at half-time and will have a scan on Monday to assess the damage
Source : PA