Liverpool ended their seven-year wait for a home league win over Manchester United on Saturday when Ryan Babel came off the bench to give them a deserved 2-1 victory over the champions.
Their victory took them to 10 points from four games, the same as Chelsea, who won 3-1 at Manchester City, despite Robinho marking his debut for City with a goal after 13 minutes.
Emmanuel Adebayor hit a hat-trick and Robin van Persie was also on target as Arsenal won 4-0 at Blackburn Rovers to go to nine points.
The gloom deepened on Tyneside where thousands of Newcastle United fans protested against absent owner Mike Ashley in the wake of the resignation of manager Kevin Keegan and then watched their team lose 2-1 to Hull City.
Marlon King got both goals as Hull claimed the club's first top-flight away win and left them in fourth place.
Jermain Defoe scored both goals as Portsmouth came from behind to beat Middlesbrough 2-1, West Bromwich Albion got their first win of the season with a 3-2 home success over West Ham United, while Fulham continued their good start by beating Bolton Wanderers 2-1.
Liverpool had lost five and drawn one of the previous six home league games against United and looked set for another defeat when new signing Dimitar Berbatov set up Tevez to fire the visitors ahead after three minutes.
A Wes Brown own goal after an error by goalkeeper Edwin van der Sar levelled before halftime and Liverpool pushed on after the break.
Their reward came 13 minutes from time when Babel got just about enough on his shot after a blunder by Ryan Giggs had allowed Dirk Kuyt to cross.
"Its been far too long now, it's not just the three points its the belief and the confidence," Liverpool's Jamie Carragher told Sky Sports News.
United manager Alex Ferguson had few complaints. "The defending was very poor for both goals but overall they were the better team," he said.
"Liverpool produced a performance of tackling and harassing us and making us make mistakes and we didn't cope with that."
Robinho, City's shock 32.5 million pounds signing from Real Madrid, made an immediate impact in the "Clash of the Cash" when he curled in a free kick after 13 minutes.
Chelsea equalised through Ricardo Carvalho three minutes later and went on to dominate.
Well-crafted second-half goals by Frank Lampard and Nicolas Anelka secured their seventh successive win over City.
Chelsea had John Terry sent off 13 minutes from time and will miss next week's clash with Manchester United - as will United's centre back Nemanja Vidic, red carded at Anfield.