Dele Alli made his return after seven weeks out with a hamstring injury and the England global duo combined to great effect for Kane's opener on 26 minutes.
"I am a little bit anxious about this change from the first half to the second half - it's only one thing, it's mental".
After Armstrong was brought down by Kyle Walker-Peters, Ward-Prowse curled a delightful free-kick over the wall and into the corner.
Here's our breakdown of this one.
Pochettino reflected in his post-match press conference: "I am a little bit anxious, this change too much from the first half to the second half is only one reason, it's mental".
"There are a lot of top sides in the league and it is going to be tough end to the season".
The win was Hasenhuttl's fourth at St. Mary's in eight Premier League games, and the Austrian will be relying on their home form in order to keep them in the division.
Substitutes: Sims (7), Long (6), Armstrong (7). Southampton's set-piece specialist then converted a free-kick in superb fashion.
It marked a quality performance from the midfielder, who has benefited greatly since the arrival of Hasenhuttl, and could find himself in Gareth Southgate's England squad due to his fine displays.
Although Solari will take charge of Madrid's La Liga game at Real Valladolid tomorrow, reports in the Spanish media say he could be dismissed next week.
"We did not take the second half seriously".
Spurs' bid for the 2016 title had also been undermined by inexperience when they dropped points in a defining fixture against rivals Chelsea, and the Argentinian said: "That is experience, through learning through history; competing in that level is not the same".
Spurs have given Hueng-min Son a rest as Lucas Moura comes in to the attack while Erik Lamela, Victor Wanyama and Fernando Llorente are among the subs.
Southampton matched the visitors during the early stages but Spurs soon stamped their authority on the game, with Dele Alli playing with the sharpness not befitting of a player who has spent over six weeks on the sidelines.
Providing the test events go to plan, Spurs will then play their opening game against either Crystal Palace or Brighton & Hove Albion.
Tottenham meanwhile don't play until March 31 when they travel to Anfield to face Liverpool.