West Ham's move from Upton Park to the former Olympic Stadium just across east London was supposed to herald the start of a successful new era.
But instead they have been beset by problems on and off the pitch and a fourth successive Premier League loss left Slaven Bilic's side languishing in the relegation zone.
The series of ugly brawls that broke out during a recent defeat by Watford raised fears about fan safety in the arena and, although there was no repeat of the crowd problems against Southampton, the lacklustre efforts of Bilic's players ensured the supporters would continue to feel uncomfortable in their new surroundings.
Austin put Southampton ahead before Tadic and James Ward-Prowse wrapped up Southampton's fourth successive win in all competitions.
West Ham were left to count the cost of a third defeat in six matches in their new abode and, with the worst defensive record in the league, they look destined for a season of struggle.
After conceding 11 goals in their last three league games, West Ham were in dire need of a vibrant performance, but it was Southampton who seized the initiative and they threatened when Ryan Bertrand picked out Pierre-Emile Hojbjerg on the edge of the penalty area.
Hojbjerg had time and space to pick his spot, but Angelo Ogbonna made up the ground with impressive speed and stuck out a leg to block the strike.
West Ham winger Michail Antonio has scored 10 headed Premier League goals in 2016 and his aerial prowess was acting as a magnet whenever his side attacked.
However, seeking out the lanky Antonio with high crosses proved a dubious strategy, with Southampton's towering centre-back Virgil van Dijk winning virtually every headed duel.
West Ham's injury problems meant Bilic was forced to deploy Havard Nordtveit out of position at right-back and the Norway midfielder was exposed when Southampton illuminated a drab first half with their 40th minute opener.
Tadic slipped a perfectly-weighted pass behind the hesitant Nordtveit and Bertrand surged away down the left flank before crossing to Austin, who guided a fine first-time finish low into the far corner.
It was the Southampton striker's fifth goal in his last four appearances and vindicated Claude Puel's decision to start him instead of Shane Long.
Puel's team had a golden opportunity to double their lead soon after half-time when Cedric Soares found the unmarked Tadic 10 yards from goal.
This time Adrian came to his side's rescue as he palmed away a Tadic strike that lacked the precision required to beat the Hammers goalkeeper.
But West Ham's creaky defence crumbled again moments later and this time Tadic didn't let them off the hook.
A total lack of communication and organisation in the Hammers rearguard proved fatal in the 62nd minute when Cheikhou Kouyate lost possession in midfield and Austin's deft pass sent Tadic clear to round Adrian and slot into the empty net.
West Ham's lone spell of pressure saw them denied a penalty when Sofiane Feghouli's shot hit Ryan Bertrand on the hand.
And it was clear the Hammers were certain to end empty-handed when Simone Zaza's shot was cleared off the line by van Dijk.
Ward-Prowse rubbed salt into West Ham's wounds as he came off the bench to drill home from close-range in stoppage-time.