Dier said, "Disappointed especially after they went down to 10 men we've got to win the game and we couldn't do that.
"They defended very well and fought for each other but that doesn't matter from our side of it. We have to be able to break them down. We weren't good enough at doing that.
"We've got to keep things in perspective. Everyone can see the progression.
"I'm enjoying learning how to play in this system."
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.
Dier added: “It didn't look that bad.
"I thought there was a Southampton player in his [Fraser Forster's] way. Goalkeepers get too much protection, especially outside the six-yard box. They can jump and use their hands which is an advantage enough."