Elliott, 16 years and 174 days old, impressed on his debut on Thursday (AEST), though he missed a golden chance early on when he turned James Milner's delivery against the crossbar.
Stuart Moore's calamitous error from Milner's 41st-minute strike gifted Liverpool the lead, however, and despite a spirited display from the home side Ki-Jana Hoever's header wrapped up the result.
There was still time for Elliott to hit the woodwork again, this time curling a strike against the bar from the edge of the box.
Only Jerome Sinclair, who made a substitute appearance against West Brom in September 2012 aged 16 years and six days, was younger when he played in a competitive fixture for the Reds.
Klopp is in no doubt former Fulham youngster Elliott proved he has a bright future at Anfield.
"At 16 [Elliott] is the kind of footballer that's really good for us. He is a brilliant boy, he takes the situation really well and wants to learn," Klopp told a news conference.
"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."
Elliott became the Premier League's youngest player when he featured for Fulham against Wolves in May, and he was one of three debutants in Liverpool's line-up along with goalkeeper Caoimhin Kelleher and forward Rhian Brewster.
"They are all good, they are all really skilled but that's why it is important to give them games like this," added Klopp, whose side will face Arsenal in the last 16.
"We will try to continue like this, so we will see. Some very good performances, very good moments, of course. MK Dons did a lot of good stuff, so it was really tricky."
While Liverpool's youngsters impressed, Klopp gave a special mention to 33-year-old Milner who, playing at left-back, made the breakthrough before teeing up Hoever for the Reds' second.
"This man is on fire, that's how it is," Klopp said. "He helps a lot and he is a real model for the young players. If he could play every day, Milly would be like this every day so that is brilliant."
Milner, meanwhile, hailed the displays of his inexperienced team-mates, saying: "It's not easy to play the way the manager wants to play so to come in and do so well shows how good they are.
"They're pushing, we see the hunger and they're improving all the time. That's what we want and hopefully we can push them and they'll keep on pushing us."