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Tottenham Hotspur will hope to become the fourth side with a 100% record when they play away to Manchester United tomorrow.
Chelsea are second on goal difference behind Liverpool after winning 2-1 away to Newcastle United. They had more than 80% possession but could not break through until the match took off in the last 20 minutes.
Eden Hazard, making his first start of the season, converted a controversial penalty after Fabian Schar was ruled to have fouled Marcos Alonso.
Chelsea were equally aggrieved by the equalising goal, claiming that Olivier Giroud was elbowed before DeAndre Yedlin crossed for Joselu to head in.
But with three minutes left Yedlin touched Alonso's shot into his own net.
"We are Chelsea and we never give up," said the outstanding Hazard.
Manager Maurizio Sarri, continuing his good start in English football, said he was surprised that Newcastle played with five defenders, which Rafael Benitez said was forced on them with so many players missing.
Watford's victory meant they have begun a season in the top tier with three successive wins for the first time, although their captain Troy Deeney felt his team had luck on their side.
"We were poor by our standards but the grit and determination and will to win has been clear to everyone," he said. "We got lucky but it gives us confidence we can be poor and still get results."
Palace manager Roy Hodgson was among those who felt Etienne Capoue could have been sent off as early as the fourth minute for the challenge on Wilfried Zaha, before setting up the opening goal for Roberto Pereyra.
"We're not happy about the foul. I've only just seen it again on television. I'd have to say we were unlucky he was not sent off," he said.
Jose Holebas added a freak second from an intended cross before Zaha halved the gap and defender Joel Ward missed a glorious chance to equalise in the last seconds of added time.
At the other end of the table promoted Fulham earned their first points in a 4-2 win over Burnley, whose Europa League exploits may be taking a toll against them.
There were five goals in the first half, two of them by Fulham's powerful Serbian forward Aleksandar Mitrovic, who then hit a post before Andre Schurrle netted the rebound for his side's late fourth.
Defeat sent Burnley, who finished seventh last season, into the bottom three above Huddersfield Town and West Ham United.