They had to battle to hold on to the three points, Alvaro Morata coming off the bench to score, before setting up a late David Luiz strike that arrived after Cesc Fabregas had been sent off as Chelsea were reduced to nine men.
Burnley's win, the club's first at Chelsea since 1971, was one of the few surprising results in the opening day of the English Premier League (EPL).
Palace fans started the season with renewed optimism amid De Boer's appointment as manager but they will be anxious by their woeful performanc, which suggests another year of struggle. We kept a good shape early in the game, but the sending off makes a big difference.
Antonio Conte was without Eden Hazard, Pedro Rodriguez and Victor Moses for the encuonter.
Burnley weren't done: Vokes made it 3-0 before half-time with a header for his second goal of the game.
It wasn't to be the end of the first half scoring.
But Liverpool came out fighting after the break, and grabbed an equaliser 10 minutes into the second half after Watford keeper Heurelho Gomes brought down Mohamed Salah in the area.
Chelsea get the defence of their Premier League title underway this afternoon as they welcome Burnley to Stamford Bridge.
Had Morata not touched a goal-bound Christensen shot over the line five minutes later, it would have been 3-2 sooner.
Burnley had to wait until their penultimate away game to win on the road last season, and Dyche smiled: "Three years ago, it was a story getting a win, the away win was a story last year, and it kills all that off".
New-look Everton had marginally the better of a disjointed opening spell with Sandro, another of their new signings, seeing an effort trundle just wide of the Stoke goal.