The champions were on the back foot after just 14 minutes when they were reduced to 10 men as Gary Cahill was shown a straight red card for a lunge on Steven Defour.
The Clarets made their numerical advantage count 10 minutes later as Vokes fired in his first of the afternoon from a Matt Lowton delivery.
Burnley extended their lead on 39 minutes when former Chelsea youngster Jack Cork set up Stephen Ward and they added a third before the break as Vokes headed home Defour's cross.
Chelsea substitute Alvaro Morata, a summer capture from Real Madrid, grabbed his first goal for the club with just over 20 minutes to play with a close-range header from a Willian cross as the Blues started to pile on the pressure.
Morata thought he had a second soon after but his strike was chalked off for offside and Chelsea's chances of fighting back appeared to be over when Cesc Fabregas received a second yellow card on 81 minutes.
Burnley picked up just one away win in the Premier League last season and the visitors had to endure a nervy finish when David Luiz fired home after latching on to Morata's flick, but the Clarets held on to record a famous victory.