The Italian was joined on the scoresheet by another new Chelsea arrival Frenchman Tiemoue Bakayoko on a satisfying night for manager Antonio Conte after a summer of much-publicised frustration in the transfer market.
"It was a good start for us. To score many goals, to win and to have a clean sheet -- I saw a lot of positive things tonight," Conte said.
The Azeri champions were never any match for their English counterparts and it took only five minutes for Pedro to open the scoring with a curling shot from the edge of the area.
The goal was a taste of things to come for the inexperienced Azeris who rarely strayed out of their half.
Deadline-day signing Zappacosta, apparently looking to cross the ball, produced a swirling screamer of a goal from some 35 metres out on the half hour. It was an eye-catching way for the Italian to open his account for the 2012 Champions League winners.
Chelsea ran riot in the second half against the Group C underdogs and equalled their best showing in the Champions League -- a 6-0 win against Maribor three years ago.
Defender Cesar Azplicueta knocked in an exquisite pass from Spanish compatriot Cesc Fabregas in the 55th minute.
Bakayoko, a second-half substitute then poked in a parried goal after a corner.
Belgian striker Michy Batshuayi added a goal in the 76th and Chelsea's sixth was an own goal by defender Maksim Medvedev six minutes later.
The Azeris, who have won their domestic championship for the last four years, were given little room and time to produce their best football.
South African striker Dino Ndlovu was frequently isolated up front and when a rare opportunity came in the final moments of the game he shot over the bar with only Thibaut Courtois in goal to beat.
The players, exhausted after chasing the game for 90 minutes, took time at the whistle to applaud their small contingent of travelling supporters in driving rain.
Chelsea will go into Sunday's Premier League derby against Arsenal brimming with confidence.
Former Arsenal midfielder Fabregas told BT Sport: "You never know if goals (difference) will count at the end of the group stage. I think it's a good performance from everyone. We are getting back to our best.
(Editing by Clare Fallon)
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