Mauricio Pochettino's side endured a disastrous start at Wembley after falling behind to Willian's early free-kick.
Harry Kane hit back through a stooping header, but the Blues were soon in front again as Willian fired home from the penalty spot.
Dele Alli netted an equaliser after the break and Spurs appeared to be in control as the closing stages of the game approached, but strikes from Eden Hazard and Nemanja Matic gave Antonio Conte's men a thrilling victory.
The north Londoners were undone by a trio of set pieces and Alderweireld is convinced he and his teammates deserved more after restricting their rivals to a handful of chances.
"We're very disappointed," he said after the game. "We deserved more and I think we were the better team.
"We came back, mentally we were strong and we played better than them.
"Two times we came back in the game so that's very good.
"There were three set pieces. They played a little bit on the counter - we knew that.
"They had four shots on goal and four goals.
"The fourth goal was an unbelievable goal so sometimes you have to have a little bit of luck in those games.
"We have to be positive - we put pressure on them and dominated the game.
"We played in their half and they were strong on the counter attack.
"But they haven't had a shot on the counter attack so we can be positive.
"We have to keep positive and try to win our games."