Chelsea had no intention of letting the minnows extend their romantic tale, pushing for a second goal that almost arrived when Willian forced Ibrahim Sehic to save.
When Conte's men doubled their lead in the 30th minute, it was laced with more than a slice of good fortune.
Taking possession deep inside his own half, Zappacosta embarked on a surge that took him clear of two Qarabag defenders and towards the visitors' area.
After looking up, seemingly to measure a cross from wide on the right flank, the Italian's whipped delivery instead caught out Sehic as it flashed into his net at the near post.
Whether he meant it or not, Zappacosta was quick to soak up the acclaim of the crowd.
Qarabag at last had a shot when Pedro Henrique brought Thibaut Courtois into action.
It was only a temporary respite and Chelsea put the result beyond doubt in the 55th minute.
Cesc Fabregas took Willian's pass and lofted a cross towards Azpilicueta, who glanced a clever header past Sehic from close-range.
Eden Hazard was cheered to the rafters when he came off the bench in the 57th minute, making just his second appearance for Chelsea this season after ankle surgery.
Willian hit the bar with a fierce strike and, with Qarabag growing more ragged by the minute, the hosts bagged their fourth in the 71st minute.
Capitalising on slack Qarabag defending, French midfielder Bakayoko grabbed his first Chelsea goal, slotting home from Hazard's cross.
Qarabag were so woeful at the back that even the shot-shy Batshuayi was able to fill his boots.
The Belgian forward struck in the 76th minute with a long-range drive, before Qarabag defender Maksim Medvedev deflected another Batshuayi effort into his own net.