Chelsea were starting to force the issue after a slow start, but the champions were reduced to 10 men in the 14th minute when captain Cahill was handed a straight red card for a late challenge on Steven Defour.
Morata did manage to give the majority of those in Stamford Bridge hope, however.
Jeremie Boga, a 20-year-old French forward, was even given a debut.
Striker Sam Vokes scored two of them by converting crosses out wide, either side of a stunning volley by left back Stephen Ward.
We must have the right experience to face this situation.
Conte said: "I prefer not to answer questions about Costa - for me it's in the past".
"Before that, we'd kept a good shape and I don't think they opened us up or made too many chances".
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Syrian military seize last Homs site from Daish
The Syrian army said it fully liberated on Saturday the city of Sukhneh, the last Islamic State (IS) stronghold in Homs province. The Syrian government still controls a pocket of territory in Deir Al Zor city, and a nearby military base.
Scaramucci: White house plotters seeking Trump exit
Many suspect that the new White House Chief of Staff, General John Kelly, will help crack down on White House leaks. Mark Meadows (R-NC), the chairman of the powerful House Freedom Caucus.
Nevertheless, David Luiz pulled back a second for Chelsea when Morata flicked on a long pass from Azpilicueta, and the Brazilian was on hand to fire past Heaton for his second goal for Chelsea since rejoining the club last season. "I haven't seen it but my staff tell me it's the correct decision", Dyche said.
Manchester City manager Pep Guardiola had some £150 million of new recruits on display at Brighton, but it took the pre-season favourites until the 70th minute to take the lead. In the next game I'm not sure whether Pedro is ready but for sure we'll have (the other) two players.
Of course, it's not just Premier League football that we'll be watching over today. I don't like this type of idea, someone sending a message.
In other results of matches also played on Saturday, Miguel Britos scored a controversial 93rd-minute equaliser for Watford as Liverpool were held 3-3 in a thrilling game at Vicarage Road. It's obvious because the linesman is on the line.
Conte is reported to be unsatisfied with the club's transfer activity and his lack of say in the recruitment process, and on the evidence of his team's early disarray at Stamford Bridge the problems extend on to the pitch.
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.
Debutant Ahmed Hegazi's first-half header earned West Bromwich Albion a 1-0 win over Bournemouth, while Mauricio Pellegrino's tenure as Southampton manager began with a 0-0 draw at home to Swansea City.