The Austrian was named as Mark Hughes' successor earlier in the day and took a watching brief at the national stadium, but for the first hour it was painful viewing.
Hasenhuttl will take over from interim Kelvin Davis, who becomes the new manager's number two, and begins the task of guiding the Saints out of the bottom three, though he will have been encouraged by the final 30 minutes as they were the better side and got a late consolation through Charlie Austin.
Before that Hugo Lloris produced a number of fine saves and they hit the woodwork three times.
The writing was on the wall as early as the third minute when Son was fed by Kieran Trippier's header and his sweet first-time volley crashed into the post.
With Hasenhuttl assessing his players, left-back Matt Targett will have earned a black mark next to his name for his lax defending that allowed Spurs to take a ninth-minute lead.
As the hosts played a corner short, Targett fatally did not match Kane's run to the near post and the England captain rammed home Christian Eriksen's cross from close range.
It was his ninth Premier League goal of the season and seventh in six games against the Saints.
With an early goal in the bank, Spurs would have fancied flexing their muscles but they were reliant on a stunning save from Lloris to keep their advantage.
The France goalkeeper somehow got his fingertips to a swerving effort from Pierre Hojbjerg and tipped it on to the post.
That gave the hosts the wake-up call they needed and they were quick to reassert their authority, with Kane forcing Alex McCarthy into a low save before the Saints goalkeeper had to smartly tip over from Son.
Pochettino's men were in no mood to let Southampton make it nervy for them and two goals in the opening 10 minutes of the second half killed the game.
After Eriksen's low free-kick was well saved by McCarthy, Moura fired home from the resulting corner at the second attempt for his fifth goal of the campaign.
Four minutes later it was three and Targett was the culprit again as Trippier dispossessed him, passed to Kane, whose low cross was easy for Son to turn in from inside the six-yard box.
Hasenhuttl will have at least been encouraged by Southampton's response to going 3-0 down as they showed plenty of spirit.
Nathan Redmond sent a swerving effort thudding into the crossbar from distance, then Trippier did well to block James Ward-Prowse's block before Lloris denied the same player with a brilliant point-blank save.
The frame of the goal was hit a third time when Mohamed Elyounoussi's header rebounded off the crossbar before they finally registered when Austin sprung the offside trap and fired home.
For Spurs, the last 30 minutes was a disappointment, but they move back into the top three after results elsewhere went in their favour.