The Londoners saw Connor Hunte sent off in the second half but held on to their lead, before grabbing a second goal in the final stages in Spain
The Blues controlled the opening stages, Lewis Baker mustering the away team's first chance of the match but fired over from range.
However, Barcelona quickly found their rhythm and were unfortunate not to grab an early lead. First in the 11th minute, Todd Kane retreated to his goal line and hacked Fernando Quesada's low shot off the line.
Then, minutes later, an enterprising run from Ebwelle down the Chelsea right had the Londoners chasing once again and Adama Traore almost capitalised as his shot slipped through Mitchell Beeney's hands. Again though, the ball was cleared off the line, this time by Alex Davey.
The panic passed and Chelsea struck following a slick piece of interplay on the edge of the Barca box. Boga's incisive pass found Kane free on the inside right of the box and he handed Feruz the chance to pass into an open goal with a clever first-time pass.
But their task was almost immediately made more difficult as Connor Hunte received a second yellow card for a foul on Adama just two minutes into the second 45.
Predictably, with a man advantage, Barca stepped up their dominance but rarely called Beeney into action. Adama's pace was a constant source of danger for the Chelsea backline and Adam Nditi was lucky not to concede a penalty for a challenge on the Mali-born wideman.
Adama's driven cross was then met by an audacious scissor kick from David Babunski that sailed out of play before Sergi Samper almost caught Beeney out with a measured chip from 25 yards.
As Barcelona continued to press for an equaliser, they were hit by a classic sucker punch as the impressive Boga weaved his way into space in the box and fired into the bottom corner, beating Ondoa and sealing their passage to the quarter-finals.