Arsenal missed out on the chance to regain top spot in the Premier League after losing 1-0 at home to London rivals Chelsea on Sunday.
Meanwhile Swansea gave new coach Francesco Guidolin victory in his first game in charge with a 2-1 win away to Everton as the Welsh club moved four points clear of the relegation zone.
At the Emirates, Arsenal were reduced to 10 men in the 19th minute when Per Mertesacker was sent off for a challenge on Chelsea striker Diego Costa, with referee Mark Clattenburg ruling the Gunners defender was the last man.
Four minutes later, Costa gave Chelsea the lead when he got on the end of a Branislav Ivanovic cross at the near post.
Arsenal’s defeat left them in third spot, three points behind shock leaders Leicester City, while Chelsea moved up into 13th place.
A hard-fought victory at Goodison Park saw Italian boss Guidlolin off to a winning start with the Swans.
Swansea went ahead though Gylfi Sigurdsson’s 17th-minute penalty but Everton were level nine minutes later when Gareth Barry’s near-post flick was turned into his own net by the Swans’ Jack Cork.
But Swansea regained the lead in the 34th minute. Left-back Neil Taylor burst clear and released Andre Ayew, whose shot deflected in off the sliding John Stones.
This was a notable win for Swansea as it meant they had recorded back-to-back league successes for the first time this season following last week’s 1-0 defeat of Watford. It was also their first league victory over Everton.
“Six points in two matches is very important for us because the table was not good,” said Guidolin. “I hope to have the possibility to go up in the table.
“In this moment I am very happy. This is the first match in the Premier League for me.”
Guidolin added: “I dreamed of working in the Premier League but I didn’t dream of winning the first match.
“I must understand what happens in this day because I am surprised. It is new for me.”
Everton manager Roberto Martinez had to make two substitutions in the first half, with the injured pair of Muhamed Besic and Kevin Mirallas now set to miss Wednesday’s League Cup semi-final second leg against Manchester City.
“We were very slow in our performance — pedestrian — but two aspects affected our momentum which was the two injuries,” said Martinez.
“The first half and second half performances were very contrasting,” the former Swansea boss added.
“We created enough chances to win the game. The fearful aspect of our performance was a real shame.”
Everton now have the worst home defensive record in the Premier League.
“We haven’t found the right way of playing at home. That is the truth,” added Martinez. “The stats reflect a really poor defensive record at home.
“We want to impress our fans, play attacking football and become a winning team but we are finding it tough to free ourselves.”