Manchester United overcame Fulham and Manchester City smashed seven goals past Norwich City in the Premier League on Saturday, but Chelsea were dealt a shock defeat at Newcastle United.
Despite a disappointing start to the season, United have now gone six games without defeat in all competitions following a 3-1 win over Fulham at Craven Cottage.
David Moyes's side went 3-0 up inside 22 minutes, with Antonio Valencia tapping in a ninth-minute opener from Wayne Rooney's square pass.
Robin van Persie's emphatic finish from Adnan Januzaj's measured through ball made it 2-0 and the Dutchman then centred for Rooney to add a third.
It can take time to settle in and it has for us, but Manchester United
usually make slow starts and get better as the season goes on
Alex Kacaniklic pulled a goal back for Fulham in the 65th minute with a shot that deflected in off Rooney, but the champions saw out a victory that elevated them to within two points of the top four.
"It's always been my team since I've taken over from Sir Alex Ferguson," said United manager Moyes.
"It can take time to settle in and it has for us, but Manchester United usually make slow starts and get better as the season goes on."
City moved into the Champions League places after bouncing back from a 2-1 loss at Chelsea last weekend by thrashing Norwich 7-0 at the Etihad Stadium.
City manager Manuel Pellegrini created a stir by dropping goalkeeper Joe Hart following a succession of blunders, but all the action came at the other end as his forwards put Norwich to the sword.
The breakthrough came in the 16th minute, the ball looping fortuitously into the net after Sergio Aguero's shot was blocked by Michael Turner and hit Bradley Johnson before crossing the line for an own goal.
Aguero teed up David Silva for the second, with Matija Nastasic heading home from a corner before Aguero found Alvaro Negredo for the fourth goal in only the 36th minute.
Yaya Toure added a fifth with a sumptuous free-kick on the hour before Aguero bludgeoned home in the 71st minute and substitute Edin Dzeko added a late seventh.
"We were very unhappy when we lost against Chelsea, but we didn't deserve to lose that match. We dominated possession," Pellegrini told the BBC.
"So we continued playing the same way. We won against Newcastle in the League Cup in mid-week and I thought today we had a really good match. Norwich did not have a chance to score, we scored seven and could have scored more."
'Deserved to lose'
Chelsea could have gone top with a victory at Newcastle, but instead they went down 2-0 on a rain-soaked afternoon at St James' Park.
John Terry twice went close for Chelsea in the first half, hitting the bar with a header from a corner and then seeing an effort cleared off the line by Davide Santon.
However, Newcastle took a 68th-minute lead when Yoan Gouffran converted Yohan Cabaye's free-kick with a diving header and Loic Remy sealed Chelsea's fate with a minute to play.
"I didn't like it. I think we deserved to lose," Chelsea manager Jose Mourinho told BT Sport.
"We had a couple of good chances to score the equaliser, but they were in the game more than us, they fought more than us and they were much more committed than us."
Chelsea's defeat saw them slip to third place, with Liverpool overtaking them on goal difference, while Newcastle climbed to ninth.
Liverpool visit leaders Arsenal in the late game.
The day also witnessed a freak goal by Stoke City goalkeeper Asmir Begovic, who scored the quickest goal of the season to date with a wind-assisted clearance that bounced over Southampton counterpart Artur Boruc after only 13 seconds.
However, Jay Rodriguez's 42nd-minute header from a raking James Ward-Prowse cross earned visitors Southampton a 1-1 draw that kept Mauricio Pochettino's side in the top five.
Second-bottom Sunderland endured a miserable day at northeast rivals Hull City, whose manager Steve Bruce enjoyed a 1-0 win over his former club thanks to a 25th-minute Carlos Cuellar own goal.
It was an afternoon tinged with disgrace for Sunderland manager Gus Poyet, who saw captain Lee Cattermole and Andrea Dossena shown straight red cards for dangerous tackles shortly before half-time.
Goals from Saido Berahino and Gareth McAuley gave West Bromwich Albion a 2-0 win at home to bottom club Crystal Palace, while West Ham United and Aston Villa drew 0-0 at Upton Park.