Elliott, who became the youngest debutant in Liverpool history, turned in an eye-catching performance on the right of a front three and hit the woodwork twice at Stadium MK.
Meanwhile, 17-year-old Dutch defender Hoever scored his first goal for the club on what was a landmark night for a host of youngsters at the club, including goalkeeper Caoimhin Kelleher and Herbie Kane.
Klopp said: "At 16 years old still and [Elliott] is this kind of footballer is really good for us. He is a brilliant boy, he takes the situation really well and wants to learn. He will always be a special player for moments, but you see overall with his defending and movements, it is not easy so that is really good.
"His left-foot is not too bad. He was a bit unlucky otherwise he would have scored. They are all good, they are all really skilled but that's why it is important to give them games like this. We will try to continue like this, so we will see.
"For Ki-Jana, I didn't play on a similar stage when I was 18 years old at this really nice stadium, I wish other clubs would have had similar dressing rooms, by the way! So in the first half he was defensively orientated. Second half he was able to get in the box and it was a wonderful goal. Wonderful cross (from Milner) absolutely."