The Scot made only three appearances in the first four months of the campaign after signing for the Reds from Hull City last summer.
But an injury to Alberto Moreno in early December opened the door for Robertson, who has started eight of the last nine games and performed impressively on each occasion.
Such was the No.26’s contribution to last Sunday’s victory over Manchester City that his name was chanted from all corners of the stadium in recognition of his exertions.
“It’s nice, really nice and very important,” Klopp said of Robertson’s recent progress.
“A lot of you [the media] were in doubt about Alberto Moreno before the season and thought we needed to bring in a new full-back, which we did with Andy Robertson. He was a nice prospect, it was clear he was a big talent and offensively skilled.
“There were a few gaps to close in his defending things so it was not the most easy time maybe for Andy, for sure. But he was quite cool in the situation – it’s not that he was a lot of times in my office and asking why I didn’t play him.
“Alberto Moreno played fantastically, apart from the Sevilla game he has played an outstanding season so far. Then he got injured and it’s very important, and for a club like Liverpool it’s unbelievably important, to have somebody.
“Andy needed maybe one game and then he was spot-on, that’s good and exactly what we need. Alberto is on the way back. That doesn’t mean anything. It only means our situation is better, that we have two players for that position. We have a lot of games and enough games for both of them.
“Andy stepped up and did a really fantastic job. It’s really nice as a manager to be part of it, to see a player [do that]. It’s quite difficult when you come to a new club and everybody expects the best of you from the first second you are in. Everybody needs time to adapt, even a football player.
“So I’m happy that he had the time, used the time and could play like he played in the last few games.”
Klopp has regularly rotated his starting line-up during the past few weeks as Liverpool navigated a hectic December and New Year fixture schedule.
And as he met the media ahead of Monday night’s Premier League clash with Swansea City, the boss was asked whether his squad is now better equipped to cope with frequent changes.
He replied: “It’s not the first time that we were aware that the depth of the squad is different to last year.
“With Joe [Gomez], we have five centre-halves but this week we had two and that happens overnight – one is injured, the other one is ill and stuff like that.
“As long as you can react it’s never a problem because it’s a normal part of the season. But you don’t want to change only when players are injured.
“We had a very intense period. These two weeks now between Everton and City, and now City and this game, it feels like a pre-season. There are so many sessions to work on different things, which we really enjoy. It’s fantastic to do things like that.
“All the other times, you have a game and a game, and recovery, and a game. And it will come up again with the Champions League hopefully and the FA Cup hopefully. So we will have enough games for the whole squad and we will use them.
“For us, there is no alternative to rotation. It felt like it was not necessary all the time but it was necessary for what’s coming up now. But nothing is decided and we still have to use this hopefully still existing physical power because now it is getting very interesting.
“Everyone fights for everything, each game becomes a final. One fights for the league, one fights for European competition, for everything. And we will be involved in all that and that’s nice.”