Subsequently forced to play with 10 men for the next 69 minutes, the Azeri champions, who had made a promising start in Baku's Olympic Stadium, were outplayed with two brilliant goals from Willian and a Cesc Fabregas penalty completing the rout.
Chelsea's win put them on top of Group C on 10 points to guarantee their place in the knockout rounds, two points clear of AS Roma, who play at Atletico Madrid later on Wednesday. Qarabag remain rooted to the foot of the table on two points.
The sending off, which enraged Qarabag and a 68,000 capacity crowd, came soon after the home side's Spanish midfielder Michel had hit the bar.
Yet everything changed in the 20th minute when Willian, trying to latch on to Hazard's slick pass, tumbled under what looked an innocuous challenge from the vastly experienced 35-year-old Sadygov, leading to a lengthy protest and an impromptu Qarabag team meeting before the penalty was taken.
The Portuguese referee Manuel De Sousa, though, had felt the defender had tugged on the Brazilian's left arm.
The visitors inevitably then took command with a dazzling four-man combination ending with a Hazard back-heel setting up a 37th-minute finish from Willian.
Chelsea manager Antonio Conte, who had started striker Alvaro Morata on the bench and brought back the out out-of-favour David Luiz, was content largely to sit on the lead after the break as the match meandered to its inevitable conclusion.
Still, another foul on Willian in the 73rd minute by Maksim Medvedev led to Fabregas scoring from a penalty that he had to convert twice after his first successful attempt had been ruled out for encroachment.
Willian, who had been involved in all four of the goals, then completed the demolition job by riding a challenge and thundering home a 20-metre strike with five minutes left.
(Reporting by Ian Chadband; Editing by Toby Davis)
