Spurs – trailing 1-0 after the first leg – came from 2-0 down on the night in Amsterdam to stun Ajax 3-2 courtesy of Lucas Moura's 96th-minute goal.
De Ligt and Hakim Ziyech put Ajax 3-0 ahead on aggregate in the first half before Lucas scored three goals after the break in sensational fashion to send Tottenham through on away goals.
It ended Ajax's fairytale run in the Champions League, having upstaged holders Real Madrid and Juventus en route to the final four and star defender De Ligt rued what could have been for the Eredivisie side.
"The referee said there was only one minute left and we knew we would have to battle through," De Ligt said via UEFA.com. "We thought we'd made it, but it wasn't to be. The game lasted five seconds too long.
"It was like the ground disappeared from under our feet. So close. We were so close. Naturally it was deathly quiet in the dressing room afterwards. But, how painful it is, we have had a fantastic journey – and we can still become champions.
"It was great to see the support our fans gave us, it made us realise how much this campaign has meant for them. If we win the title I will feel very proud, and it will help soften the blow of standing with one foot in the Champions League final with only seconds to go.
"Spurs played with a great deal of intensity, and that caused us a lot of problems, particularly in the second half. We couldn't really relieve the pressure they put us under."