The Clarets made their numerical advantage count 10 minutes later as Vokes fired in his first of the afternoon from a Matt Lowton delivery.
And though Alvaro Morata and David Luiz would net in the second half to spare Chelsea some embarrassment, Cesc Fabregas' late second booking further reduced their chances of completing an improbable comeback.
There had been a change of guard over the summer with John Terry's departure with Cahill taking the armband, but his league debut as permanent captain lasted all of 14 minutes.
And Conte says he isn't anxious about his squad depth.
Tom Ince missed an early chance to give the Terriers the lead when he sent a bouncing effort from Tommy Smith's cross straight at keeper Wayne Hennessey.More news: Flood Situation Worsens In Parts Of North East, 4 Dead In Assam
On Saturday they were missing Eden Hazard and Pedro through injury as well as suspended Victor Moses.
Chelsea very almost completed a dramatic comeback in the second half, with goals from substitute Alvaro Morata and David Luiz, but a red card for Cesc Fabregas meant that their attempt to recover was carried out with nine men, and Burnley were resolute enough to secure the win.
But Rooney ended a scrappy first half when he headed in to mark his Goodison homecoming in the best possible style.
Burnley went for the fourth goal but the post kept out a 90th minute freekick as the defending champions sought an equaliser.
In the 3pm kick-offs, champions Chelsea were predicted to start their campaign with an easy win over Burnley at Stamford Bridge, only for Antonio Conte's men to find themselves 3-0 down at half time. Our performances against the big teams were pretty good a year ago, barring Chelsea away.
But they held out, fully earning a result that sees them match their tally of Premier League away wins in the whole of last season.