Liverpool, leading 2-0 from the first leg in England last week, was rocked by two first-half goals from the Spanish side. However, three second-half strikes secured their third Champions League final in five seasons under the German.
"It feels like the first in 20 (years)," Klopp, who led Borussia Dortmund to the final of the competition in 2013, told BT Sport. "It's outstanding because we obviously made it a little tricky for ourselves, but we knew these kinds of things could happen."
Liverpool head to the final in Paris to face either Real Madrid or Manchester City, who play their second leg in Madrid on Wednesday with City holding a slender 4-3 lead on aggregate.
Klopp's side were shocked by Villarreal's aggressive man-marking in the first half which took them out of their own pressing game, and the coach conceded that he struggled to find highlights from the opening 45 minutes to show his players.
"We knew what was wrong because it was obvious, but we didn't have a situation to show them where we got it right," Klopp explained.
"I said (to my staff) 'find one where do it well and we can show it, and we come in and they said 'no, we don't have it.'"
The introduction of Colombian winger Luis Diaz at the break seemed to liven up Liverpool's attack as Fabinho, Diaz and Sadio Mane scored the goals that got them into the final.
"Obviously Luis, what a player he is, but it's not about who came on, it was about how we started to play," Klopp explained.
The 54-year-old coach said he had no preference for who his team would face in the final on May 28, saying "whoever it will be, it will be massive".
For opposite number Unai Emery it was the end of an epic Champions League campaign for Villarreal, who knocked out Juventus and Bayern Munich en route to the semifinals.
"Tonight we showed everybody we are a good team as well, we can have chances... but the difference in two matches, in the first leg and the second leg was their excellence," the Spaniard told BT Sport.