Conte has struggled to hide his frustrations over Chelsea's failure to strengthen their squad, with Morata, Antonio Rudiger, Tiemoue Bakayoko and back-up goalkeeper Willy Caballero their only new signings.
Sam Vokes scored his second goal to make it 3-0 for the visitors in the 43rd minute.
"The first half pleased me because although Chelsea went to 10 men it was not easy".
The pendulum had swung in the wake of Cahill's dismissal and in the 24th minute Vokes propelled Burnley ahead when he steered a cross by Matthew Lowton into the net.
Morata did manage to give the majority of those in Stamford Bridge hope, however.
Huddersfield Town made a dream start to their debut season in the Premier League as they beat Crystal Palace 3-nil away.
The home side were back in the match in the 69th minute when Morata headed a Willian cross into the bottom corner to register his first goal for the club, although the champions still had a lot to do to turn the game around.
Conte, looking exhausted and disgruntled, said he preferred not to talk about the referee who also sent off Cesc Fabregas in the 82nd minute after the Spaniard earned a second yellow card for an ill-timed tackle on former Chelsea man Jack Cork.
As for Liverpool, they conceded a 94th minute goal at Watford in a 3-3 draw.
"It was a fantastic three points and the referee, I just want to say, was fantastic today".
Bakayoko was absent through injury against Burnley, along with star man Eden Hazard, while striker Diego Costa has been frozen out after being deemed surplus to requirements.
"This result kills off the away win story for this season but we're not getting carried away", said Dyche.
Who will be the first managerial casualty of the new season? "I am ready to fight with these players". Why do I have to send a message?
Leroy Sane, Raheem Sterling, and Bernardo Silva " attacking midfielders costing a combined $175 million " came off the bench to help City clinch a 2-0 win at promoted side Brighton. Conte has made it clear that the club must bring in fresh faces before the transfer window closes on the 31st August.
Wayne Rooney marked his top-flight return to Everton after 13 years by scoring the victor against Stoke and his 199th goal in the Premier League.