Mauricio Pochettino's side could not go on to find a winner, though, as Spurs remain two points outside the Champions League places and could end the gameweek even further adrift should Liverpool beat Swansea City on Monday night.
Southampton, meanwhile, see their winless streak stretch to 11 games and remain in the relegation zone, but the result will be a satisfactory one for the under-fire Mauricio Pellegrino as his side closed to within one point of safety.
The Saints had dropped into the bottom three courtesy of Stoke City's win over Huddersfield Town on Saturday, but they started the better and brought the first save from Michel Vorm - in for the unwell Hugo Lloris - after only 11 seconds when Manolo Gabbiadini turned and shot from 20 yards.
It was down the left flank that Southampton were getting the most joy during the opening exchanges, but both Gabbiadini and Pierre-Emile Hojbjerg failed to make the most of separate low crosses into the box when they could not get anything on the ball.
A similar delivery did provide a deserved opener for Southampton soon enough, though, with Ryan Bertrand bursting down the left flank before sliding a cross into the box which Sanchez inadvertently turned into his own net when trying to cut the ball out.
Spurs were quick to hit back, though, and a snapshot from Eric Dier struck the outside of the post before Kane restored parity just three minutes after falling behind.
The England international brought up Premier League goal number 99 by shaking off the marking of Jack Stephens before climbing highest to meet a Ben Davies corner inside the area and plant a header past Alex McCarthy from close range.
Tottenham almost made it two in the space of just a minute when Dele Alli collected the ball inside the box moments later, but he could not bring it under his spell and find an unmarked Kane in the middle before Stephens came in with a last-ditch challenge.
Southampton were still carrying a threat of their own, though, and they came close to regaining the lead in the 26th minute when Dusan Tadic teed up Mario Lemina, who drilled an effort over the crossbar from the edge of the area.
Spurs almost took the lead on the half-hour mark when a drilled delivery from Davies picked out Moussa Sissoko, but the winger could not react in time to steer it on target with the goal gaping.
Southampton soon hit back as the two sides continued to trade chances, and the hosts should have been ahead when a free kick from James Ward-Prowse found Stephens unmarked inside the area, only for the defender to steer his header narrowly wide of the post.
Dier then drilled a low effort towards goal which McCarthy got down well to save before Vorm was called into action at the other end to keep out Lemina's curling strike from the edge of the box.
It took just two minutes for the first sight of goal in the second half too as Ward-Prowse forced a save from Vorm, but the chances were not as free-flowing in the early stages of the second 45 minutes.
Alli did almost put Spurs in front shortly after the hour mark when he drilled a low shot narrowly wide, but again Southampton were quick to respond in kind as Gabbiadini's bouncing low drive needed to be saved by Vorm.
Tottenham's pressure did begin to grow as time ticked away, though, and the visitors squandered a big chance to take all three points when Erik Lamela placed his finish wide from point-blank range after being picked out by Sissoko.
Pellegrino turned to 17-year-old striker Michael Obafemi in a bid to turn the game back in his side's favour late on, and the youngster almost made it a fairytale debut when he had the chance to latch on to a cross into the box, only to get his finish all wrong.
The hosts had another clear opening in the closing stages when Sofiane Boufal was teed up inside the area, but the winger took too long over his finish and Sanchez was able to make an important block just in front of the keeper.
It looked as though Spurs might punish Southampton for those missed opportunities in the final minute of normal time when Kane was slid in down the left channel, but with his Premier League century waiting he dragged his finish uncharacteristically wide of the far post.
That proved to be the final big chance of the contest as Southampton held out for a point, leaving Spurs playing catch-up in the top-four race ahead of games against Manchester United, Liverpool and Arsenal in their next three league outings.
Southampton, meanwhile, have now gone 11 top-flight games without a win for the first time since 1998 and remain in the relegation zone as a result, albeit now only one point from safety.
SOUTHAMPTON (4-2-3-1): McCarthy; Cedric, Stephens, Hoedt, Bertrand; Romeu, Lemina (Davis, 80'), Ward-Prowse, Hojbjerg (Boufal, 65'), Tadic; Gabbiadini (Obafemi, 82')
TOTTENHAM (4-2-3-1): Vorm; Aurier (Trippier, 72'), Sanchez, Vertonghen, Davies; Dier, Dembele (Wanyama, 85'), Sissoko, Alli, Son (Lamela, 70'); Kane