Reds keeper Adam Bogdan stopped Junior Stanislas’s close-range shot with his legs as Bournemouth started brightly.
Liverpool, who drew their three opening games under Klopp, led when Clyne pounced after Adam Smith cleared Joao Carlos Teixeira’s flick off the line.
Bogdan saved from Matt Ritchie in injury time as Liverpool held on.
The Reds, who have won the League Cup on a record eight occasions, move into the quarter-finals for the second successive season.
Klopp will take his team to Premier League rivals Southampton in the last eight.
Liverpool’s attack has been severely depleted by injury, with Christian Benteke joining Daniel Sturridge on the sidelines after being ruled out of the tie on Wednesday morning.
Their absence, both through knee injuries, meant Klopp placed his faith in £29m summer signing Roberto Firmino and 20-year-old Belgium international Divock Origi.
Ex-Dortmund boss Klopp said Firmino was “the best player in the Bundesliga” for a short time last season — and the Brazil international responded with a standout performance against the Cherries.
Firmino’s vision created the winning goal, picking out Teixeira whose cheeky backheel could only be cleared into the path of Clyne.
And former Lille striker Origi, who is yet to score for the Reds since arriving at Anfield this season following a loan year in France, can also take credit for his part in the winner.
He had few opportunities to break his scoring duck, but his pace pulled apart the Cherries defence, allowing Firmino to pick out Teixeira. —BBC