Chelsea's recent woes continued with a 1-1 draw at Crystal Palace, meaning they have taken just two points from four games over the festive period.
Defending champions City returned to winning ways at Leicester last week after a long lean spell and picked up three more points against Julen Lopetegui's West Ham at the Etihad.
Pep Guardiola's men took the lead in the 10th minute when Vladimir Coufal deflected Savinho's cross past goalkeeper Alphonse Areola.
Savinho also had a hand in the second goal, shortly before half-time, beating his man twice before crossing to Haaland, who lost Max Kilman and headed home.
The same players combined for City's third in the 55th minute, before Phil Foden joined the party for their fourth. The home side were denied a clean sheet by Niclas Fuellkrug.
West Ham appeared to have turned the corner last month but they have now conceded nine goals in two games following a 5-0 mauling at home by Premier League leaders Liverpool.
'Angriest in career'
In the early kick-off at the Tottenham Hotspur Stadium, Newcastle came from behind to clinch a fifth straight Premier League win and heap more misery on injury-hit Spurs.
The home side went ahead in the fourth minute through Dominic Solanke.
Newcastle levelled two minutes later when Antony Gordon arrowed an effort into the bottom corner but Postecoglou stood in disbelief on the touchline looking at referee Andrew Madley after Joelinton intercepted Lucas Bergvall's pass with his hand in the build-up.
VAR deemed Joelinton's arm to be in a natural position and the contact to be accidental, which provoked a furious reaction from Tottenham's bench and they were up in arms midway through the first half when the already-booked Dan Burn avoided a second yellow card for handball by the centre circle.
In-form Alexander Isak scored what turned out to be the winner towards the end of the first half.
The victory leaves Newcastle in fifth spot, which may yet prove enough to secure a place in the Champions League next season, but Spurs have slumped to 12th in the table after five defeats in seven matches.
Postecoglou said he was proud of his players but was the "angriest I think I have ever been in my career".
"I think it's clear," said the Australian. "Now whether people agree with me or not whether it wasn't handball or it was accidental, I'm just not interested in any of that discussion.
"I know what everyone wants me to say, but all I'll say is that on any other day, on a fair and even playing ground, we would have won that game. Simple as that."
Newcastle boss Eddie Howe admitted the ball hit Joelinton's arm but said he was not in control of the referee's decision.
"I believe his arm is down in a natural position and the referee and VAR have followed the rules and protocols that are in place," he said. "That's why it is given."
Enzo Maresca's Chelsea were closing in on Liverpool just two weeks ago but are now nine points behind the Reds, having played two games more.
The Italian made six changes for the match at Selhurst Park and his side took the lead when Jadon Sancho drove down the left and found Cole Palmer, who sliced a low shot into the bottom right corner.
The visitors enjoyed the bulk of possession but they could not capitalise and Jean-Philippe Mateta levelled in the 82nd minute.
Aston Villa beat relegation-threatened Leicester 2-1 at home with goals from Ross Barkley and Leon Bailey.
Brentford recorded a rare win on the road, smashing bottom side Southampton 5-0, and Bournemouth beat Everton 1-0.
Second-placed Arsenal are in action at Brighton in the 1730 GMT kick-off while Premier League leaders Liverpool host crisis-hit Manchester United on Sunday.
