Spurs lost 4-2 at Emirates Stadium on Sunday but banished those memories with goals from Harry Kane, Lucas Moura and Son Heung-min inside 55 minutes to lift them above Chelsea to third in the Premier League.
Southampton, who gave Spurs boss Mauricio Pochettino his first job in English football, sacked Mark Hughes following their Saturday draw with Manchester United and Hasenhuttl - confirmed as the Welshman's successor earlier on Wednesday - was in the Wembley stands to watch the club's eighth defeat of the top-flight season.
The visitors - coached by Kelvin Davis in this match - hit the woodwork three times before Charlie Austin netted a stoppage-time consolation, but three soft goals had already killed off a Southampton side who are now only kept off the foot of the table by goal difference.
Son almost had the home side in front after only three minutes, rattling the right-hand upright on the volley after Kieran Trippier nodded into his path.
Southampton were not so fortunate six minutes later, when Kane drifted into the six-yard box to touch home a low Christian Eriksen delivery from the left.
The visitors created a chance from nowhere as Pierre-Emile Hojbjerg's beautiful 25-yard strike found the woodwork with the slightest of touches from the fingertips of a fully stretched Hugo Lloris.
The lively Son forced Alex McCarthy to tip over his crossbar and the Southampton goalkeeper was kept busy early in the second half, too.
He was beaten for a second time in the 51st minute when Lucas buried a shot from inside the box after his first effort bounced back off the hand of Jack Stephens.
A poor Matt Targett clearance then allowed Kane to square a pass for Son to net the third from close range, before a late Saints flourish saw Nathan Redmond hit the crossbar and Lloris forced into a brilliant double save from Hojbjerg and James Ward-Prowse.
Southampton hit the woodwork one more time through Mohamed Elyounoussi and finally found the net with a neat finish from Austin after Steven Davis' assist, long after the match had been lost.
What it means: Spurs up to third
Pleasingly for Spurs, the result lifts them back above rivals Arsenal and beyond Chelsea - who lost at Wolves - into third place. Pochettino's men sit eight points adrift of leaders Manchester City and six behind Liverpool.
Southampton, who are 18th, are now winless in 13 matches in all competitions, their longest such run since hitting the same number between May and October 1998.
Kane in the neck
England striker Kane has proven to be a real thorn in the side for Southampton in recent meetings between the teams, and it was no different here as he registered a goal and an assist.
The England striker scored a hat-trick in this fixture last season and has now scored seven times in his last six league games against Saints.
Hasenhuttl may wish he had stayed away on Wednesday, as Southampton turned in the kind of error-strewn display that has blighted their season so far, conceding three soft goals and committing too many sloppy mistakes.
Saints had let in 14 goals in their past five away matches in Premier League prior to the trip to Wembley and another feeble performance at the back should put sorting that defence close to the top of the Austrian's to-do list.
Key Opta Facts:
- Tottenham have now won each of their last 10 home Premier League games against sides starting the day in the relegation zone, netting 23 goals whilst conceding just three.
- Spurs have now won 11 and lost four of their 15 league games this season, meaning only Manchester United (17 in 2012-13) and Bolton Wanderers (18 in 2011-12) have opened a Premier League season playing more matches before their first draw.
- Southampton have earned just nine points from their opening 15 games this season; just three of the previous 18 teams to have nine points or fewer from their first 15 games of a Premier League season have avoided relegation.
- Eriksen has provided at least one assist in each of his last three Premier League games (four total), the first time he has done so within a single campaign since April 2016.
- Tottenham's Oliver Skipp (18y 79d) is the youngest player to debut for the club since Gareth Bale against Manchester United in August 2007 (18y 40d).
Tottenham visit Leicester City on Saturday ahead of a crucial Champions League clash with Barcelona at Camp Nou next Tuesday, while Hasenhuttl's reign begins with a winnable trip to Cardiff City, who sit two points above them.