Chelsea recovered from the shock of blowing a two-goal lead against Arsenal as Marcos Alonso's late strike sealed a 3-2 win, while Harry Kane ended his August hoodoo in Tottenham's 3-1 victory over Fulham on Saturday.
Maurizio Sarri's side raced into a 2-0 lead in the London derby before Arsenal drew level in a thrilling first half at Stamford Bridge.
Pedro swept home with a cool finish in the ninth minute after woeful Arsenal defending and Alvaro Morata crashed in the second 11 minutes later.
Although Arsenal were rocky at the back, they were dangerous going forward and Henrikh Mkhitaryan got one back with a low strike after 37th minute.
Alex Iwobi nabbed Arsenal's equaliser with a clinical effort in the 44th minute.
But Chelsea bounced back from that blow, dominating the second half before Spanish defender Alonso finished off substitute Eden Hazard's pass in the 81st minute.
Chelsea's second successive win under new boss Sarri lifted them to the top of the Premier League with a 100 percent record, while Arsenal remain without a point under their new manager Unai Emery after back-to-back losses.
Bournemouth also have maximum points from their opening two games as Eddie Howe's men came from behind to win 2-1 at West Ham.
Everton's �50-million signing Richarlison continued to quieten queries of his price tag with the winner as Southampton were defeated 2-1 at Goodison Park, while Leicester saw off Wolves 2-0 despite a red card for Jamie Vardy.
After becoming the first Premier League side since the introduction of the summer transfer window to fail to sign a player, Spurs were dealt another blow off the field in midweek with the news their new 62,000-capacity stadium won't be ready until at least October.
Mauricio Pochettino's men will therefore host another two home fixtures against Cardiff and Liverpool at Wembley.
Lucas Moura curled home his first Premier League goal to give Spurs the lead, but Aleksandar Mitrovic deservedly headed Fulham level early in the second half.
- World Cup stars shine -
Tottenham may still be suffering a World Cup hangover with the majority of their stars enjoying barely a week of pre-season training after their efforts in Russia.
But it was two of England's World Cup heroes who produced the goods as Kieran Trippier curled home a wonderful free-kick before Kane finally ended his wait to score in the opening month of the season at the 15th attempt in the Premier League.
"We dominated the first 30 minutes and created a lot of chances, but in the Premier League if you drop your level it can let in the other team," Pochettino said.
In contrast to Spurs, West Ham splurged nearly �100 million in the transfer market to back new manager Manuel Pellegrini but still have no points to show for it after a second-half surrender to Bournemouth.
Marko Arnautovic's penalty gave the hosts a half-time lead. Two goals in six minutes turned the game on its head, though, as Callum Wilson and Steve Cook made it two wins from two for the Cherries.
Everton's revolution under Marco Silva gained more momentum at Goodison with Richarlison scoring his third goal in two games.
A well-worked free-kick freed up Theo Walcott to open the scoring before the Brazilian headed home a Walcott cross to double Everton's advantage.
Wolves shared the points in a four-goal thriller with Everton last weekend, but despite much hype and expectation after a summer of heavy investment, they remain winless after losing 2-0 at 10-man Leicester.
A spectacular own goal from Matt Doherty and a deflected James Maddison strike put the Foxes in command.
However, victory came at a cost for Leicester as England international Vardy was shown a straight red card 20 minutes from time for a late lunge on Doherty and now faces a three-game ban.
Cardiff and Newcastle both registered their first point of the season in a drab 0-0 draw in the Welsh capital in a missed opportunity for both sides.