Sam Vokes capitalised on the numerical advantage and Chelsea's poor defending, scoring either side of a goal from team mate Stephen Ward, to send Burnley into the halftime break a worthy if unexpected 3-0 up.
Chelsea looked in good form at the start, in spite of the absence through injury of key players, until Cahill lunged clumsily at Steven Defour's shin after losing the ball high up the pitch and referee Craig Pawson was prompt with a red card.
Chelsea pulled one back midway through the second half as record summer signing Alvaro Morata came off the bench to head home but any sign of a comeback vanished when Cesc Fabregas was dismissed for a second booking in the 81st minute. Their reward? To lead the top-flight table for the first time in more than 70 years, ahead of Manchester City who defeated Brighton 2-0 in the evening kick-off.
Azpilicueta backed Conte's repeated assertion that Chelsea need more experienced options, claiming that the current situation will not even benefit the youngsters who stand to be given more opportunities on the pitch.
The Blues are also preparing for a return to the Champions League this season after their title triumph last term.
"I want the deal with Atletico resolved this month", he added.
He considered trying to sign Drinkwater along with Kante last summer and have retained an interest ever since.
Despite the fact that Chelsea's Premier League season didn't get off to the best of starts following their defeat to Burnley, Antonio Conte isn't panicking just yet because there's an abundance of talent in the champions' squad.
The Blues have expressed an interest in right-backs Joao Cancelo and Serge Aurier, who are expected to leave Valencia and Paris Saint-Germain. He's done it consistently over the last few years.
Antonio Rudiger has called on his Chelsea team-mates to make amends for their shock loss against Burnley by bouncing back against title rivals Tottenham next weekend.