Ayegbeni Yakubu hit two goals on his debut and Arsenal's hapless defence added two own goals as Blackburn registered its first win of the season to ramp up the pressure on Gunners manager Arsene Wenger.
Swansea romped past West Bromwich Albion with goals from Scott Sinclair, Leroy Lita and Nathan Dyer, while Joey Barton, Alejandro Faurlin and D. J. Campbell gave Queens Park Rangers a 3-0 win at Wolverhampton Wanderers.
Norwich completed the trio of victories for the promoted clubs, Anthony Pilkington and Bradley Johnson getting first-half goals in a 2-1 win at Bolton. Martin Petrov cut the deficit with a 64th-minute penalty.
In other matches, Newcastle recovered to draw 1-1 at Aston Villa to preserve both sides' unbeaten starts to the season and Everton scored two late goals to beat Wigan 3-1.
Arsenal's troubled start to the season deepened despite goals from Gervinho in the 10th minute and Mikel Arteta in the 34th twice giving Arsene Wenger's team the lead.
Yakubu equalized in the 25th and an own goal from Alexander Song just after halftime pulled Blackburn level again as Arsenal's shaky defence self-destructed under the pouring rain at Ewood Park.
"It is terrible, it is just not good enough," Wenger said. "Of course we are very frustrated, the spirit in the team is quite willing but, if you look at the number of goals we have conceded, that is not good enough."
Yakubu added a third in the 59th and Laurent Koscielny became the second Arsenal player to put into his own net following Martin Olsson's burst down the right in the 69th.
"We had to press, we had to fight, and we played on the counterattack and I think our second-half performance shows what we've got in that dressing room which is 100 per cent team spirit," Kean said. "The boys are really working hard for each other and I'm delighted."
Several hundred Blackburn fans had demonstrated their dissatisfaction with Kean outside the ground before the match started, but the home side's win meant it was Wenger who was fielding questions about his job.
"Speculation is part of modern life, what can I do about that?" Wenger said. "My future is focusing on what I do well and that means giving my best for the club. I have no influence on the speculation."
Meanwhile, Norwich manager Paul Lambert made six changes to the side beaten by a single goal at home to West Bromwich Albion last weekend and was delighted with his side's 2-1 win at Bolton.
"I've never had a problem with confidence," Lambert said. "They seem to bounce back whether they draw or get beaten, and that is testament to the way they played today."
Bolton slipped to a fourth straight defeat after Pilkington's close-range finish in the 37th and Johnson's header in the 42nd gave the Canaries a two-goal lead at halftime.
Bolton manager Owen Coyle was disappointed both goals came from set pieces — a traditional strong suit for his team.
"To concede the two goals from set plays certainly wasn't good enough. We're all designated markers and we haven't done well enough in that department," he said. "It's not as if someone has beaten two or three players and hit it into the top corner — this was self-inflicted."
Ivan Klasnic complicated Bolton's task when he was sent off just after the break for an apparent headbutt on Marc Tierney — an action described by Coyle as "an act of folly."
QPR secured a 3-0 win over Wolves after new captain Joey Barton scored his first goal since moving from Newcastle.
Barton steered the ball home in the eighth minute and Faurlin smashed the second from some 20 yards on the half-volley just two minutes later as QPR made a quick start.
The visitors continued to press, and substitute Campbell got his first goal for the club by sliding in Armand Traore's pass in the 87th.
QPR manager Neil Warnock fielded all five of the permanent signings recruited since Tony Fernandes' takeover of the London club, and it was a convincing performance by the new-look team.
In Wales, Sinclair ended Swansea's 374-minute wait for a topflight goal as he coolly slotted home a 14th-minute penalty kick against West Brom.
Lita nodded home his first goal for the club 10 minutes later before Dyer secured the points shortly after the break, with Brendan Rodgers' side also keeping a third straight clean sheet at home.
Elsewhere, Newcastle striker Leon Best scored his third goal of the season to ensure a share of the spoils at Villa Park
"Leon has great belief in himself," Newcastle manager Alan Pardew said. "He gets a start, he gets a goal, and he keeps the shirt, it's as simple as that in my book."
Gabriel Agbonlahor gave Villa the lead when his volley from Barry Bannan's left-footed cross hit Fabricio Coloccini and rebounded back to the England striker to score from close range in the 13th minute.
The point pushed Newcastle into fourth place provisionally.
At Goodison Park, substitute Apostolos Vellios scored in the 84th minute and Royston Drenthe also came off the bench to score in injury time to complete an impressive comeback for Everton.
Wigan went ahead in the 31st after Franco di Santo was given too much time to collect a near-post corner from Ben Watson. Di Santo scored with an angled shot which deflected off Tony Hibbert and into the far corner.
Everton equalized within 97 seconds through two headers — Phil Jagielka finally scoring from close range after Tim Cahill's first attempt rebounded off the crossbar. It was only Jagielka's third goal in four years for the Toffees.
Fixtures continue Sunday with league leader Manchester United hosting third-place Chelsea and second-place Manchester City visiting bottom club Fulham.
Two other clubs will be looking to break into the top four when Liverpool travel to Tottenham Hotspur and Stoke City visits Sunderland.