Klopp said he was "heartbroken" to learn Diogo Jota, who he signed in 2020, had died at the age of 28.
Jota died alongside Andre after their vehicle veered off a motorway in northern Spain before bursting into flames.
Klopp, who is Red Bull's "head of global soccer", persuaded the Liverpool board to pay £45 million for Diogo Jota after he impressed at Wolves.
"This is a moment where I struggle!" Klopp posted on Instagram.
"There must be a bigger purpose! But I can't see it! I'm heartbroken to hear about the passing of Diogo and his brother Andre.
"Diogo was a not only a fantastic player, but also a great friend, a loving and caring husband and father!
"We will miss you so much! All my prayers, thoughts and power to Rute, the kids, the family, the friends and everyone who loved them! Rest in peace - Love J."
Diogo Jota was labelled an "exceptional player, exceptional boy," by Klopp early on in the player's spell at Liverpool.
Jota took little time to deliver some return on Liverpool's investment, becoming the first player in the club's history to score in his first four home games.
He was a key part of Klopp's side that won the League Cup and FA Cup in 2021/22, narrowly missing out on the Premier League and Champions League for what would have been an unprecedented quadruple.
He was part of last season's Premier League-winning squad.
Liverpool said in a statement: "Liverpool Football Club are devastated by the tragic passing of Diogo Jota.
"The club have been informed the 28-year-old has passed away following a road traffic accident in Spain along with his brother, Andre.
"Liverpool FC will be making no further comment at this time and request the privacy of Diogo and Andre's family, friends, teammates and club staff is respected as they try to come to terms with an unimaginable loss.
"We will continue to provide them with our full support."
Liverpool teammate Stefan Bajcetic, who spent last season on loan at Las Palmas, was the first current player from the Reds to release a comment, posting an image in training with Jota on Instagram with the caption: "You will be remembered my friend."
Darwin Nunez shared an emotional post online for his "good companion" and teammate Jota.
"There are no words of comfort for so much pain," Nunez said.
"I will always remember you with your smile, as a good companion on and off the field. I send all my strength to his family, from where he is I'm sure he will always be with you, especially his wife and three children."
Basketball legend and Liverpool shareholder LeBron James posted via his social media: "My prayers goes out to his loved ones during this time! May you all be guided and protected! YNWA (You'll Never Walk Alone) JOTA!!"
"Devastating news about Diogo Jota and his brother Andre this morning," former Liverpool defender and Sky Sports pundit Jamie Carragher posted on X.
"Thoughts are with every one of their family and friends, especially his wife Rute and their three lovely kids."
"RIP Diogo," club legend Steven Gerrard said. "Condolences to his family and friends during this incredibly sad time."
Former Liverpool player and manager Sir Kenny Dalglish said that he felt "helpless" upon learning the news.
"Football is not important at this sad time," Dalglish said on X.
"You feel helpless, knowing there's so little we can do to ease the pain for his wife of just two weeks, his three beautiful children.
"Thinking and praying for all their loved ones after the devastating loss of both Diogo and Andre. YNWA."
Liverpool midfielder Harvey Elliott paid tribute to Jota on his Instagram page: "Diogo, I can't believe you're gone. It's hard to put into words how much this hurts us.
"You were an incredible footballer, the kind that made a difference every single time you stepped on the field. You were humble, hardworking, kind, and always there for anyone that needed anything.
"I feel so lucky to have shared the pitch with you, shared memories and moments I'll carry with me forever. You'll always be part of this team, part of the Liverpool family. We will never, ever forget you.
"My heart goes out to your whole family.
"Love you brother. Rest easy Diogo and Andre. You'll Never Walk Alone."
"Words cannot describe how heartbroken and devastated we are... Your smile, your love for the game will never be forgotten," current teammate Dominic Szoboszlai said.
"We will miss you so much, but you will stay with us forever, on and off the pitch."
Former Liverpool stars Roberto Firmino and Sadio Mane also both paid tribute to Jota, with Mane posting a picture of the pair on his Instagram account.
"It was a privilege and an honour to have known and lived such special moments alongside you," Firmino said.
"When words fail, there are only inexpressible groans."
In a lengthy statement posted via Liverpool's official website, current manager Arne Slot delivered an emotional message.
"All I have are feelings that I know so many people will share about a person and a player we loved dearly and a family we care so much about," he said.
"My first thoughts are not those of a football manager. They are of a father, a son, a brother and an uncle and they belong to the family of Diogo and Andre Silva who have experienced such an unimaginable loss.
"My message to them is very clear - you will never walk alone. The players, the staff, the supporters of Liverpool Football Club are all with you and from what I have seen today, the same can be said of the wider family of football.
"My condolences go to Diogo’s wife, Rute, their three beautiful children and to the parents of Diogo and Andre Silva.
"When the time is right, we will celebrate Diogo Jota, we will remember his goals and we will sing his song. For the time being, we will remember him as a unique human being and mourn his loss. He will never be forgotten.
"His name is Diogo."