The Dutch giants were on track for the final, taking a 2-0 second-leg lead to go 3-0 up on aggregate in Amsterdam.
However, Lucas Moura's second-half hat-trick – completed with a dramatic 96th-minute winner – sent Spurs into the decider against Liverpool in incredible fashion.
Ten Hag rued the way the loss unfolded, but said his Ajax team would have to move on as they continue their Eredivisie title challenge.
"I told the players that first of all it's very hard to find the right words for a moment like that, and that football can be cruel," he told a news conference.
"This is the cruel side of football that we have to endure tonight, but we'll have to go on.
"We can't do that in a couple of hours or a couple of days, I can't ask that of the players, but if you're so close and the Champions League is so unique, and the players all realise that a lot, you need time to get over that, but we'll have to.
"We'll have to get back on track before Sunday [against Utrecht] and we'll have to start with that on time. You cannot wait for too long or we won't be ready for Sunday."
Despite first-half goals from Matthijs de Ligt and Hakim Ziyech, Ten Hag said he warned his players about a possible Spurs comeback.
The former defender lamented his side's failure to make the most of their chances on the break, while rueing Ziyech hitting the woodwork.
"We were so close and we deserved it. Giving away the lead was careless. I calmly told the players that it was not over yet. You saw that Spurs still believed," Ten Hag said.
"You have to keep going, but it became increasingly difficult. Spurs started to play more opportunistically and we just didn't play well on the counter-attack.
"If we had scored instead of hitting the post, we would have ended the match."