Antonio Conte says Tottenham will have to "improve a lot" and it "won't be easy to fight for something this season" after they were held by a Southampton side who played for more than 50 minutes with 10 men.
Spurs fell behind at St Mary's when James Ward-Prowse cut across a bouncing ball to volley a swerving shot past Hugo Lloris for a superb opener.
But the game turned when Mohammed Salisu fouled Son Heung-min in the box to earn a second booking in 12 first-half minutes and allow Harry Kane to level from the penalty spot.
Conte became the first manager in Tottenham's history to remain unbeaten in his first seven league games, but said: "My demands are very high because I am used to being competitive and to win - and, for sure, for me there is a big disappointment when we don't win.
"I understand also the situation. Now for sure we have to work a lot and it won't be easy to fight for something this season.
"We need to improve a lot. For sure it was an opportunity that we missed. When you have this type of situation you have to try to exploit, to try to get three points."
Spurs twice had the ball in the net after the break, with Kane seeing a strike ruled out by VAR for a narrow offside and Southampton goalkeeper Fraser Forster spared own-goal blushes when it was judged he was fouled by Matt Doherty.
Tottenham felt aggrieved by the decision on Forster's foul but, in truth, they did not do enough after the break to pull and stretch a patched-up Saints back four, who repeatedly limited the visitors to crosses from wide areas.
Southampton boss Hasenhuttl punched the air in delight at the final whistle and said there was "no chance to be more proud than today".
Conte has now overseen four Tottenham wins and three draws in the league, and the likes of Pierre Emile-Hojbjerg explained before kick-off that the boss has called on players to raise their game technically, physically and in how they live as professionals away from the training ground.
Their effort could not be questioned in the second half at St Mary's, with Doherty denied by a Forster save, Son seeing a header held, and substitute Bryan Gil having a stinging shot tipped over in injury time.
Forster made 10 saves in all - the most by a keeper to not end on the losing side in the Premier League this season.
If anything, Spurs simply took too long to build momentum with their one-man advantage, and Southampton - beaten only by Wolves at home in the league this season - ground out a point to move to 13th in the table.
Hasenhuttl made six changes from the Boxing Day win at West Ham and his side's grit and resilience in the second period was typified by Kyle Walker-Peters, who leapt to glance a header away from the lurking Kane before immediately taking a Ben Davies shot flush in the face.
Spurs saw 53% of first-half possession, registering four shots. Those stats rose to 76% and 18 attempts in the second period - pointing to the significance of Salisu's dismissal.
The Ghanaian defender had appeared on edge in the first half, recording four fouls in 39 minutes as Southampton looked to play high up the pitch and harass Spurs into errors. His dismissal was Southampton's 11th since Hasenhuttl took charge in 2018, with only Arsenal having a higher tally, 13, in the same time frame.
The sharp Kane lashed his penalty high into the net and later came incredibly close to staying onside before applying a finish to a Harry Winks pass six minutes after the break.
But with the likes of the impressive Ward-Prowse winning all of his duels on the day, Southampton kept a solid shape to force Spurs wide. Conte's side delivered 28 second-half crosses compared with three in the first 45 minutes and had little to show for their approach.
And Southampton could have won it in injury time when Armando Broja robbed Tottenham defender Eric Dier before firing over, sparing Spurs further frustration on a day when they could have taken more.
