Heung min-Son fired the hosts into the lead but Southampton rallied and kept their hopes of a Europa League place alive courtesy of a Steven Davis brace.
Spurs were still fighting for the title a week ago but have now taken just two points from their last three games.
Sunday's defeat to Ronald Koeman's side was preceded by draws against West Brom and Chelsea, which handed Leicester the title, and Dier found both results difficult to take.
Tottenham vs Southampton: Player Ratings
"It's been a tough couple of weeks with the draws against West Brom and Chelsea," he told Spurs TV.
"That hurt us mentally and coming into this game [against Southampton] we were on quite a low to be honest.
"We tried to pick ourselves up but we haven't done as well as we've been doing all season so everyone's disappointed."
Dier also lamented Tottenham's inability to hang onto a lead against Southampton, which has become a worrying trend for the north Londoners in recent weeks.
"It's that same story where we've gone ahead and ended up losing or drawing.
"It's something we're going to have to look at and fix next season.
"We'll pick ourselves up. There's still one game left so hopefully we can finish well," he insisted.
Spurs travel to Newcastle on the final day of the season knowing a point is likely to be enough to secure second place ahead of Arsenal, barring a huge swing in goal difference.
