Liverpool booked their place in the Carabao Cup fourth round for the first time in three years as they sealed a routine 2-0 win over MK Dons.
The Premier League leaders, who will face Arsenal at Anfield in the next round, took advantage of a goalkeeping howler on the stroke of half-time as Stuart Moore spilled James Milner's shot into his net (41).
By that stage, Conor McGrandles had missed a glorious chance to open the scoring for the Sky Bet League One side when he lifted his shot over from six yards before Harvey Elliott - on his Liverpool debut - struck the bar from close range.
But despite Paul Tisdale's side giving a very good account of themselves in front of 28,521 spectators, a record attendance for the Dons, Ki-Jana Hoever sealed the win with his first Liverpool goal, heading in Milner's cross (59).
MK Dons: Moore (5), Williams (7), Walsh (6), Poole (6), Gilbey (6), Brittain (6), Kasumu (6), McGrandles (7), Dickenson (6), Bowery (7), Nombe (7).
Subs: Martin (6), Houghton (6), Boateng (6).
Liverpool: Kelleher (7), Hoever (7), Gomez (7), Lovren (6), Milner (7), Keita (6), Oxlade-Chamberlain (7), Lallana (6), Jones (8), Elliott (8), Brewster (7).
Subs: Kane (n/a), Chirivella (6), Van Den Berg (n/a).
Man of the match: Curtis Jones.
How youthful Reds set up Arsenal tie
Liverpool had lost their last four League Cup games - their worst ever run of defeats in the competition - but despite 11 changes to the side that beat Chelsea 2-1 on Sunday, Jurgen Klopp's mixture of youth and experience continued their fine start to the campaign.
The eight-time League Cup winners might have fallen behind when the impressive Sam Nombe escaped Dejan Lovren on his first start of the season before picking out McGrandles six yards out, but he was unable to keep his shot on target.
Elliott, who demonstrated just why Liverpool were so desperate to prise him away from Fulham during a thrilling first appearance in a Reds shirt, should have opened his account when his shot from Milner's low cross crashed off the underside of the bar.
It did not affect the 16-year-old's performance as he escaped two MK Dons defenders down the right midway through the first period to pick out Milner at the far post, but the left-back for the night could only direct his header wide.
Paul Tisdale made five changes to the side that were beaten 1-0 at home by Southend at the weekend.
Stuart Moore replaced Lee Nicholls in goal, while George Williams, Jordan Bowery, Alex Gilbey and Brendan Dickenson came in for Hiram Boateng, Dean Lewington, Russell Martin and Jordan Houghton.
Jurgen Klopp made 11 changes as Caoimhin Kelleher, Rhian Brewster and Harvey Elliott made their senior debuts. Dejan Lovren made his first start of the season while Naby Keita returned from injury.
The hosts looked like they would go in at the break level, but a lapse in concentration gifted Klopp's side the lead as Milner cut inside onto his right foot to shoot straight at the body of Moore, but the ball squirmed out of the goalkeeper's grasp and he was unable to prevent it from crossing the line.
There was always a measure of control about Liverpool's performance in the second half, but Jordan Bowery came within inches of levelling just shy of the hour-mark as substitute Russell Martin flicked Alex Gibley's cross into the striker's path but he could only poke his effort against the post.
The miss proved costly as Hoever met Milner's cross to stretch Liverpool's lead with his first senior goal to put the European champions firmly in control.
MK Dons ended strongly with Bowery brilliantly denied by Reds 'keeper Caoimhin Kelleher before George Williams saw his effort cleared off the line by Lovren.
There was still time for Elliott to strike the woodwork again in stoppage time, as he was denied the goal his performance merited, but a third would have been cruel on the spirited hosts.
How did Liverpool's youngsters fare?
Klopp put his faith in five players under the age of 21 in his starting line-up in Milton Keynes, and none of them let him down during a highly professional performance.
Caoimhin Kelleher overcame an anxious moment in the opening period when he was fortunate not to be penalised for handling just outside his penalty area, but the young goalkeeper ensured Liverpool avoided a nervous final 10 minutes by preserving their 2-0 lead, superbly keeping out Jordan Bowery from close range.
Ki-Jana Hoever had already announced himself to the senior side with his ability to turn defence into attack when he was thrown into Liverpool's FA Cup tie at Wolves earlier this year, and he showed even greater maturity on Wednesday as he met Milner's cross to cap a disciplined performance with a goal.
Curtis Jones was the pick of the bunch with his direct running and willingness to provide a constant outlet down the left on only his second appearance for the club.
Co-commentator Don Goodman named the 18-year-old his man of the match: "He's been neat and tidy. He's been superb, popping up all over the pitch. He's creative and certainly one to keep an eye on."
Jones was certainly not alone in impressing as Harvey Elliott was twice denied a debut goal by the crossbar.
The former Fulham academy player might have done better with his close-range effort in the first half, but his head never dropped and he showcased his confidence in stoppage time as he cut onto his weaker foot only to see his effort again come back off the bar.
Much is expected of Rhian Brewster, and while he was closely shackled by the Dons' defence, he showed glimmers of real promise as he narrowly failed to connect with one Milner cross.
Milner happy with 'solid performance'
"It's always tough when you have a mixture of guys who haven't played many minutes and a group of youngsters, but you could see the style of football we wanted to play.
"We lost our way a little bit in the second half, and didn't keep the ball as well, but I thought it was a pretty solid performance.
"When you look at how good our squad is, it's not easy for the youngsters to get to the level they need to be , but they've shown they want to get better and better. Everyone is pushing, and that's what the senior boys need competition for places."
What the managers said
MK Dons boss Paul Tisdale: "I'm really proud of the players and saw a lot of effort. I was so pleased with how we applied ourselves but I'm frustrated as it could've been better.
"I'm disappointed that we didn't find a goal as I feel we deserved it. We missed a good chance in the opening five minutes, and we might have scored a goal in the final 10 minutes to put them under pressure, but it wasn't to be."
Liverpool boss Jurgen Klopp: "This experience is very important for the young boys. We had around 80 per cent of the ball but with their 20 per cent they had four very good chances. They had these moments in the final 10 minutes, so it's less a physical thing and more about concentration.
"Caoimhin's a wonderful goalkeeper and I'm pleased that he could show that tonight. I saw a lot of good and exciting performances, moments here and there, and that's what we already know about the boys.
"The young boys are confident, but it's important that life gives you a little knock, or a push. It's not that easy when you play with the real men."
Man of the match: Curtis Jones
Jones was the outstanding performer in his second Liverpool appearance, never hiding from the ball as he stretched the MK Dons' defence throughout.
Jones told Sky Sports: "I was pleased to show the fans what I could do and it was great to play alongside some of the senior players. I'm hoping for a few more opportunities along the way."
Liverpool's youth and experience in stats
At 16 years and 174 days old, Harvey Elliott became the youngest ever player to start a competitive match for Liverpool, and the second youngest to play overall after Jerome Sinclair against West Brom in September 2012 (16y 6d).
James Milner scored his first goal in the League Cup since the 2010 final for Aston Villa against Manchester United, 9 years and 209 days ago. This was Milner's 20th appearance in the competition since then.
Liverpool travel to face Sheffield United in the Premier League on Saturday - kick-off is at 12.30pm. MK Dons return to League One action on the same day with a trip to Sunderland at 3pm.