Norway captain Brede Hangeland scored a first-half goal to hand Fulham, who escaped relegation on the final day of last season, an upset 1-0 home win over Arsenal in the English Premier League on Saturday.
Liverpool captain Steven Gerrard scored four minutes into stoppage time to cap a thrilling 2-1 comeback victory over Middlesbrough at Anfield that gives his side maximum points after two rounds.
A header by substitute Michael Owen earned Newcastle United a 1-0 home win over Bolton Wanderers in his first game of the season while Tottenham Hotspur went down to a shock 2-1 home defeat by Sunderland, debutant Djibril Cisse netting the winner.
Fulham, who lost 2-1 at promoted Hull City on the opening day of the season, felt their surprise victory proved they are worthy of remaining among the elite.
"I think we were a bit unlucky last week... When we play our best football like we did in the first half it's possible to beat any team," Texas-born defender Hangeland told Sky Sports.
Fulham manager Roy Hodgson added: "Today we've beaten one of the best teams in the league to show we may have the pedigree to stay in this league.
Arsenal manager Arsene Wenger said: "Credit to them, they were always sharp, motivated, defended very well...In the first half in midfield we were dominated."
Middlesbrough substitute Mido, who netted in their 2-1 victory over his former club Spurs last weekend, put 'Boro ahead with a long-range strike 20 minutes from time on Merseyside.
Liverpool kept their fans on tenterhooks until Jamie Carragher's shot was deflected by Middlesbrough defender Emanuel Pogatetz into his own net in the 86th minute.
Gerrard then produced a late angled strike to earn the home side a dramatic last-gasp victory.
Former Liverpool striker Cisse, who has joined Sunderland on loan from Olympique Marseille, sent ambitious Tottenham slumping to their second successive defeat when he nodded the winner from Daryl Murphy's left-wing cross in the 82nd minute.
"We had threat. Sometimes you're only as good as your strikers. Lucky for us, one of our substitutions worked out and Cisse scored," Sunderland manager Roy Keane told Sky Sports.
Sunderland had gone ahead at White Hart Lane with a brilliant 25-metre drive from Kieran Richardson before Spurs, who left want-away Bulgaria striker Dimitar Berbatov out of their squad, equalised through Jermaine Jenas in the 73rd.
Newcastle were lucky not to fall behind on the hour, goalkeeper Shay Given saving Kevin Nolan's penalty.
Manager Kevin Keegan had intended to give Owen, returning from a calf injury, only a few minutes near the end but his hand was forced by injuries to forwards Damien Duff and Obafemi Martins during the match.
"Then [Owen] did what he does better than anybody in the world. It was a great cross by Geremi and this guy [Owen] does deliver quality," Keegan said.
Mali striker Mamady Sidibe struck four minutes into stoppage time as promoted Stoke City beat visitors Aston Villa 3-2, while Everton won 2-1 at West Bromwich Albion and Blackburn Rovers were held 1-1 at home by Hull.
Chelsea, one of the title favourites, play at Wigan Athletic on Sunday and champions Manchester United visit Portsmouth on Monday.