The Scot has deputised in the absence of the injured Alberto Moreno excellently in recent weeks and produced another excellent display against the Cherries as he continually got forward.
He helped to force the opening goal as his run distracted Bournemouth's defence, allowing Philippe Coutinho to work his magic, open up a space and drill a low shot into the bottom corner.
Dejan Lovren's header and Mo Salah's 20th goal of the season had the Reds 3-0 up by half-time before Roberto Firmino killed any faint hopes of a comeback past the hour mark with a header.
Yet it was left-back Robertson - starting his fourth successive Premier League match - who drew more praise than any of Liverpool's other individuals from his manager.
Robertson a "fantastic football player"
Asked about whether he is grasping his chance after being made to wait, Klopp defiantly replied: "He didn't have to wait. He had to train and to learn. That's a different thing. Only because everyone is asking, it looks like he has to wait."
The German insisted he is "happy" about Robertson's performances, declaring that he "made a big step" at the Vitality Stadium.
"The first two games [he played] were all good but Robbo can do much better and he knows that," Klopp explained, insisting that such expected improvement is "normal" because players "have to gain rhythm" and "need to get more confident."
He added: "Today was a game when he really made the difference. The first pass he played in the back for Roberto, Roberto couldn't get it but that was the idea and he did really well. He's a fantastic football player so I am happy for him."
Reds have "so many things to play for"
Liverpool remain 18 points behind league leaders Manchester City, but Klopp remains confident that his team have plenty yet to contest between now and May though their ambitions of a title challenge have long been scuppered.
Klopp - on whether the title race is already over given City's 11-point lead over second-placed Manchester United - said: "I don't have a crystal ball. It looks like it. I have to say, [they have got] all my respect."
He praised City's "fight" and explained that while they are "not flying in the moment" they "really work hard for it" and claimed their commanding lead at the summit of the top flight is "deserved."
Klopp said that if they struggle like "nobody has struggled before", at least a team with City's quality, then "somebody should be there" to capitalise upon it - but warned: "I don't see it in the moment."
"If I would have stopped working hard in the moment when I cannot be champion anymore in the league then I would not be here," he said. "We have so many things to play for. I enjoy the season in this moment."
He noted that it is "really hard" and "really intense", insisting it is "unbelievable" how many points "you will probably need at the end of the season to get anything."
Klopp continued: "Most of the time we've played our football, it is really nice to watch and successful. We play next Friday against Arsenal, that’s a really hard one as well, but I'm looking forward to it."
Liverpool needed 76 points just to finish fourth last season and Klopp admits it is likely his side must produce a similar sum come the end of this term.
"I don't think it will be much lower. Everybody is winning up there very often so you need to carry on all the time and stay on track," he admitted. "I don't know how many points. It doesn't make it but more likely when I say it, but a lot of teams are in good shape and City win all their games. It's difficult to get them."