Replacing Michy Batshuayi with almost an hour gone, Morata entered the game with his team 3-0 down after a nightmarish first half.
"Winning away from home on the first days quietens down some stories - we're written off at the start of every season".
Referee Craig Pawson, right on the spot, instantly brandished a red card.
Conte returned to his mantra from last season that the team needed to work hard to produce results and he pointed out that Chelsea had finished their last three games without all 11 of their players on the pitch. Cahill challenged for the ball with one foot, but his studs were showing, prompting Pawson to dismiss the Chelsea captain. Andreas Christensen made his debut as a sub for debutant Jeremie Boga, and Burnley took advantage of an unfamiliar Chelsea back three.
Stephen Ward compounded Chelsea's misery just after the half-hour mark when he controlled a long pass and beat Antonio Rudiger to lash home from a tight angle on the left.
Morata made the flawless impact as he pulled a goal back for Chelsea in the 68th minute as he headed home Willian's cross.
His new side may have fallen to defeat at Stamford Bridge, but Morata remains positive. It's obvious because the linesman is on the line.
A sense of injustice spurred on Chelsea's riposte, which only grew when Cesc Fabregas became the second player to walk, for a second yellow card, less than ten minutes after substitute Alvaro Morata had reduced the arrears to 3-1.
The result is also the first time Chelsea has lost an opening league game of the season since 1998.
Chelsea down to 9 men.
Conte will have some thinking to do, along with the board, as both Cahill and Fabregas will be missing for next week's match with Tottenham Hotspur.
"The sending off of Cahill was a key moment, because to play the rest of the game with 10 and then nine men is not easy", the Italian told the BBC as he reflected on more red cards for his side, coming a week after Pedro was sent off in the Community Shield against Arsenal at Wembley Stadium.