Antonio Conte's side moved up to second with the win, a position they held after Manchester United were held to a 0-0 home draw by Southampton in the evening kick-off.
Swansea were trailing 1-0 at Watford until the 86th minute but goals from Jordan Ayew and Luciano Narsingh turned the game around to lift Calvalhal's new side off the bottom of the table.
Mohamed Salah took his tally of Premier League goals for the season to 17 with a double as Liverpool came from behind to beat Leicester, who had led at Anfield through Jamie Vardy's early goal.
The Reds remain fourth in the table, four points ahead of Tottenham and Arsenal in fifth and sixth.
Chelsea 5-0 Stoke
Watford 1-2 Swansea
Liverpool 2-1 Leicester
Manchester United 0-0 Southampton
Ryan Fraser also scored twice, for Bournemouth, who inflicted a first defeat for Sam Allardyce as Everton manager.
There were no goals for any of the side's promoted from the Championship last season as Huddersfield and Burnley finished goalless, as did Newcastle v Brighton.
Bournemouth 2-1 Everton
Huddersfield 0-0 Burnley
Newcastle 0-0 Brighton
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In Scotland, Rangers held Celtic to a 0-0 draw in the Old Firm derby, but Brendan Rodgers' team remain unbeaten against their fierce rivals since 2012 and seven points above second-placed Aberdeen at the top of the Scottish Premiership table.
Derby moved up to second in the Championship with a 2-1 victory at Ipswich. County are seven points behind leaders Wolves, who face third-placed Bristol City at 17:30 on Saturday.
There were also big wins for two of the sides in the bottom three, with Birmingham beating Leeds 1-0 and Bolton claiming a victory by the same scoreline at Sheffield United. THINK YOUR FRIEND WOULD BE INTRESTED? SHARE THIS STORY USING ANY OF THE SHARE BUTTON BELOW ⬇ PLACE YOUR TEXT ADVERT BELOW:>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>