Not only have the Saints secured victory in Europe, they have won their first game of the season thanks to Austin and substitute Rodriguez.
It was a perfect start for the Saints as they went ahead in the fifth minute, but it did not happen without a bit of controversy.
Sparta's Costa was penalised for a handball, but there was not much that the defender could do about it as the ball bounced off his leg and onto his arm during a tackle on Austin.
Initially, referee Manuel Grafe awarded a free kick on the edge of the box, but he switched to a penalty after discussing the incident with his linesman.
Austin lined up to take it, despite Dusan Tadic seemingly being the regular penalty taker, and the Serbian was not best pleased as he protested with his teammate, but the striker went ahead and slotted the ball into the back of the net.
Virgil van Dijk came close to scoring a second for Southampton a few minutes later when James Ward-Prowse's corner kick found him in the centre of the box, but the defender's header glanced over the crossbar.
Southampton had a comfortable first half, and Austin staked a claim for regular first-team inclusion when he nabbed a second goal one minute after he had another disallowed for being offside.
Cuco Martina made it for the striker as he delivered a pin-pointed cross into the box for Austin to head past the keeper and double the hosts' lead.
Towards the end of the first half, Sparta had an opportunity to pull a goal back when Julis stormed into the six-yard box, but his final touch let him down as the ball soared over the top.
The Saints took some time to get going in the second half, but Tadic looked eager for a goal of his own when he exchanged passes with Pierre-Emile Hojbjerg before cutting inside and sweeping a left-footed shot towards goal but keeper Tomas Koubek comfortably saved.
The visitors only created a handful of goalscoring opportunities during the match, one of which came from a free kick, which was delivered into the box by Vaclav Kadlec, but Tadic got there first to clear the danger.
After a lengthy lull, Ales Cermak tried to spark it into life with a solo run, and after his shot was blocked, David Lafata was there for the follow-up, but his half-volley strike was directed straight at Fraser Forster.
In the closing stages of the match, the Saints had a penalty denied, despite Maya Yoshida getting a nudge in the back.
It was a comfortable night's work for the Premier League club as Rodriguez wrapped it up by directing the ball past the keeper after latching on to Long's pass.
Southampton top the group following Inter Milan's 2-0 defeat to Hapoel Be'er Sheva.