Jose Mourinho endured his first Premier League loss since returning to Chelsea at Everton on Saturday, while Manchester City was frustrated in a draw at Stoke.
For Arsenal, Manchester United and Tottenham, however, there were routine victories as the league swung back into action for the first time since the transfer window closed in the international break.
Samuel Eto'o failed to make an immediate scoring impact on his Chelsea debut as Steven Naismith gave Everton a 1-0 win.
Naismith headed into the net from close range in first-half stoppage time on his 27th birthday after Nikica Jelavic nodded the ball across to the forward.
While it was Roberto Martinez's first win as Everton manager after three draws, Chelsea has two wins from four league matches since Mourinho — the self-styled "Special One" — returned after six years.
"It was a special game, a special opposition and a very good performance," Martinez said.
Mourinho saw things differently.
"We didn't deserve to lose because we were the best team because we played the best football because we dominated the whole game," he said.
Arsenal reaped some early reward from its decision to spend a club-record 50 million euros ($66 million) on Mesut Ozil on transfer deadline day two weeks ago. The Germany playmaker set up Olivier Giroud for the opener before in-form Aaron Ramsey scored twice in a 3-1 win at Sunderland. A third straight win maintained Arsenal's resurgence after a demoralizing opening-day defeat to Aston Villa.
"You've seen what he is all about, slipping those balls in," Ramsey said of Ozil, who was substituted after 80 minutes on his debut. "He has one of highest assist rates in Europe and he is going to create a lot of opportunities for us."
Man City settles for draw
Manchester City's mixed start to the season continued by scraping a 0-0 draw at Stoke. United was also far from fluent in a 2-0 home win over Crystal Palace that turned on a contentious decision to send off Kagisho Dikgacoi in the 43rd minute for a foul on Ashley Young.
Despite the challenge coming outside the area, a penalty was awarded — and converted by Robin van Persie — before Wayne Rooney curled in an 81st-minute free kick to hand David Moyes a first success at Old Trafford. Rooney was playing with a thick headband to cover a deep cut to the forehead sustained in a training-ground incident two weeks ago.
Tottenham began life without Gareth Bale, who joined Real Madrid two weeks ago, by outclassing Norwich 2-0 with a double from Gylfi Sigurdsson, who was set up for his first goal by new signing Christian Eriksen. Arsenal and Tottenham are level on nine points while the Manchester rivals both are on 7.
Liverpool protects its perfect start to the season when it plays Swansea on Monday.
Elsewhere Saturday, France playmaker Hatem Ben Arfa scored one and set up another as Newcastle beat Aston Villa 2-1 away for its second victory in a row. West Bromwich Albion claimed its first point of the season by scoring in injury time for a 1-1 draw at Fulham and Hull drew 1-1 with Cardiff in a match between two promoted clubs.
While United was thankful for a refereeing decision to turn its game against Palace, Arsenal also benefited from a debatable call at the Stadium of Light when the score was 2-1.
United States striker Jozy Altidore beat Bacary Sagna to the ball and prodded home a finish, only for the referee to bring play back for an earlier foul against Arsenal.
It was too much for combustible Sunderland manager Paolo Di Canio, who was sent from the dugout in injury time.
"He came to me and said if you keep going with your manner, I send you up to the stand," Di Canio said. "I said to him, 'If you want to complete a perfect job, you send me off' and that is what he did."