Considering the club's problems with their strikers of late - Christian Benteke not deemed 100 per cent fit to join Daniel Sturridge and Danny Ings on the sidelines - it required someone else to step up.
Clyne would not have been many people's choice as the match-winner but after three successive draws Klopp was more than happy to take it after making nine changes from the side which drew at home to Southampton on Sunday, with only Clyne and Divock Origi surviving.
The German handed first starts to academy graduates Joao Teixeira, Cameron Brannagan and Connor Randall, making his debut and all fared well.
But the fact none of starting XI had scored this season and only two, Dejan Lovren and Joe Allen, had ever found the net for the club would have given Bournemouth, who made six changes, encouragement.
And considering the Liverpool defence's long-held lack of confidence the result could have been different had they taken the lead after five minutes.
Junior Stanislas cut inside Lovren, leaving the Croatia international on his backside, but gave Adam Bogdan the chance to save with his legs.
The Bournemouth midfielder had two other chances in the first half with a close-range header and free-kick but on both occasions the Liverpool goalkeeper was more than a match.
Even with the regular generosity presented by their opponents' defence visitors cannot afford to squander too many opportunities at Anfield and with £29million summer signing Roberto Firmino starting to find his feet in a central, attacking midfield role Liverpool began to dominate.
The Brazil international, who twice threatened with shots from outside the area, played a significant part in their 17th-minute goal as his slide pass picked out the perfectly-timed run of Teixeira.
Showing the cockiness of youth the 22-year-old Portuguese's clever backheel beat Adam Federici and although Adam Smith got back to clear off the line Clyne, who had charged forward from left-back, fired home the weak clearance.
Firmino should have doubled the lead after half-time when Randall's pressure forced Marc Pugh to concede possession to the Brazilian but his angled shot flew wide of the far post.
It may have been because it was only the League Cup, possibly because Bournemouth had offered little threat or even because there has been something of a sea-change at Anfield, but there was none of the previous anxiety a 1-0 lead would have held heading into the last 20 minutes.
The Kop appeared much more relaxed and that seemed to filter down to the players with Teixeira's confidence still on a high as he forced Federici into a good save from his 20-yard free-kick.
But with no alarms at the other end Liverpool managed to hold on to a lead for only the fourth time this season to offer yet more progress for Klopp.
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