Chelsea created the first chance after just four minutes when Islam Feruz brought a smart stop from Connor Ripley.
A few minutes later the away side nearly took the lead when dangerman Lucas Piazon, who was very impressive throughout, shot narrowly wide.
Boro then started to get a foothold in the match and Cameron Park found himself in space in the right hand channel but saw his left-foot curler go wide.
Luke Williams was next to test the keeper after being set up by good work from Matthew Waters and Ryan Brobbel.
However, it was the Stamford Bridge boys who took the lead on 22 minutes through a stunning Billy Clifford free-kick. His right-foot shot beat Ripley as the ball flashed past him from fully 25 yards.
Boro tried to hit back when Lewis Sirrell broke down the left and played the ball across to Brobbel but his shot was well blocked by Saville.
Chelsea almost doubled their lead on 31 minutes but George Saville's shot was well saved by Ripley.
Boro recovered and came close to levelling when first Williams saw his shot blocked then a follow-up from Sirrell was saved.
The miss proved costly as Chelsea broke quickly and extended their lead through Lewis Baker, who finished with a low shot from inside the penalty area.
Half-time: Boro U21s 0 Chelsea 2
The second half started disastrously for Boro, Alexander Davey scoring Chelsea's third a minute after the restart with a simple close-range finish following a neat build-up.
Boro showed the resilience which has been a common theme this season and pulled a goal back on 49 minutes with a fine finish rom Ryan Brobbel.
The comeback was well and truly on after 65 minutes when the ball pinballed its way around the Chelsea box from a corner and Kyle Oliver got the decisive touch to grab Boro's second.
Jamie Clapham's side were now camped in the Chelsea half and were nearly caught out by a quick break but Clifford hit the ball wide.
Boro captain Matty Dolan chanced his arm but his 20-yard howitzer just cleared the upright. This proved to be his last action of the game, as he limped off to be replaced by Jordan Jones.
The game was swinging end to end now and Nathan Ake chose the wrong ball when through for Chelsea and Piazon's shot was saved.
It was oh-so-close to a Boro third when Cameron Park found some space on the right hand side but his effort was smothered by the keeper.
Both sides needed to win this game and hope Wolves didn't beat Manchester City to have a chance of qualifying for the elite group in the next stage.
City were in the lead for a long spell before Wolves struck twice late on. There was still time for drama when City won a last-minute penalty.
Score and Chelsea went through, but they didn't, as Wolves held on to progress with Liverpool.
The way the groups are set up, Boro and Chelsea are now likely to meet again in the next phase of the competition, which will get underway in the New Year.
Boro: Ripley, Bennett, Sirrell (Kneeshaw 79), Dolan (Capt) (Jones 75), Burgess, Oliver, Brobbel, Park, Waters (Edwards 89), Williams, Reach. Unused subs: Bland (gk); Fowler.
Chelsea: Blackman, Christensen, Davey, Ince, Bangura (Ashton 75), Ake, Baker, Clifford, Feruz (Phillip 62), Saville (Capt), Piazon (Pappoe 89). Unused subs: Beeney (gk); Musonda.