The champions slumped to a 3-2 opening day defeat against unfancied Burnley, shipping three goals in the first half after Cahill was sent off for a unsafe tackle.
Playing in the top flight for the first time in 45 years, Huddersfield beat Crystal Palace 3-0 away as record signing Steve Mounie scored twice in a dream start for the promoted club.
His goal galvanised his team and the Stamford Bridge crowd, and 19 minutes later the former Real Madrid striker set up David Luiz with a glancing header for Chelsea's second.
Conte was proud of the second-half effort, but acknowledged Chelsea lost their heads after Cahill saw red, a period in which Fabregas got his first booking for sarcastically applauding referee Craig Pawson, who came under pressure from the home players and supporters.
Sam Vokes scored in the 24th minute, and then the 43rd, whilst Stephen Ward made it three for three in the 39th.
Conte said he did not want to answer questions about Costa, but tellingly conceded: "For me it's in the past".
Eden Hazard, Pedro Rodriguez, and Tiemoue Bakayoko are injured, striker Diego Costa is estranged, and Conte has a threadbare squad after selling midfield anchorman Nemanja Matic to rival Manchester United and loaning out numerous club's youngsters.
Two of the favourites for this season's Premier League title race, Liverpool and Chelsea, both failed to win on Saturday.
"This result kills off the away win story for this season but we're not getting carried away", said Dyche.
"We controlled the counter-attacks and we didn't concede one chance", Guardiola told BBC Sport. They're heavily tipped to end the season rock bottom, but the Terriers could give the first indication today that they're prepared to spring a surprise. I don't like this type of idea, someone sending a message. As for the transfer market, the club is trying to do the best.
Rooney marked his first league start for Everton since rejoining his boyhood team from Manchester United by netting in a 1-0 win over Stoke at Goodison Park.