Having embarked on a £250 million ($280 million) spending spree in the summer transfer window, Chelsea are expected to deliver on their new owners' eye-watering investment.
The early returns had been mixed as the Blues crashed to defeats against Leeds and Southampton.
When Michail Antonio gave West Ham a shock 62nd minute lead with a close range finish, the critics were sharpening their knives at Stamford Bridge.
But Ben Chilwell equalised in the 76th minute, just three minutes after the England left back replaced Marc Cucurella.
Havertz then cleverly finished off Chilwell's cross to complete Chelsea's escape in the 88th minute.
There was still more drama as Maxwel Cornet appeared to equalise in stoppage time, only for the goal to be controversially disallowed by VAR for Jarrod Bowen's foul on Edouard Mendy.
Thomas Tuchel's side are up to fifth place after their third win in six games, while West Ham boss David Moyes was livid with the officials for denying his team a point.
"The goalkeeper comes to take it, and actually fumbles it out of his hands five or six yards, so he could never recover it. Then he acted as if he had a shoulder injury. I'm amazed that VAR sent the referee for him to see it," Moyes said.
"But I thought even if he goes to the TV there's no way he's overturning this, because this is a goal. It was a ridiculously bad decision. The sad thing is this is the level of the weak refereeing at the moment."
Erling Haaland has been remarkably prolific since his £51 million move from Borussia Dortmund in the summer, and he struck again in City's 1-1 draw with Aston Villa.
The Norwaegian striker's 10th goal in his first six Premier League games meant he became the joint fastest player to reach double figures, sharing the record with Micky Quinn, who hit the same tally for Coventry in 1992.
But Pep Guardiola's side were punished for failing to build on their lead as Bailey's blistering finish left the champions frustrated.
- Spectacular Pickford -
In a goalless Merseyside derby at Goodison Park, there was more VAR intrigue as Conor Coady thought he had put Everton ahead against Liverpool with a close range finish from Neal Maupay's shot in the second half.
But Coady's goal was correctly disallowed for offside after a VAR check.
There could easily have been goals aplenty but for a man of the match display from Everton keeper Jordan Pickford and some fine stops from his Liverpool counterpart Alisson Becker.
Everton are now without a victory in their first six games this term, while Liverpool's momentum was checked after winning their previous two matches.
Unbeaten Tottenham made it four wins from six games with a 2-1 victory against Fulham in north London.
Pierre-Emile Hojbjerg broke the deadlock for Tottenham in the 40th minute, sweeping home from Richarlison's pass.
Harry Kane increased Tottenham's lead in the 74th minute from Ryan Sessegnon's cross to become the third highest goal-scorer in Premier League history, with 188 goals taking him past Andy Cole.
Only Alan Shearer (260) and Wayne Rooney (208) are now above Kane in the scoring chart.
Fulham's Aleksandar Mitrovic curled home in the 82nd minute, but Tottenham held firm to win.
- Delight for United and Brighton -
Antony announced himself to Manchester United fans with a debut goal against Arsenal, which was added to by a double from Marcus Rashford as the hosts won 3-1.
England international Bukayo Saka scored an initial equaliser for the Gunners, but they were easily carved open again by United and handed their first loss of the season.
Brighton also enjoyed a positive Sunday, beating Leicester 5-2 in a thrilling encounter at the Amex stadium.
The Foxes took an early lead through Kelechi Iheanacho, but quick replies from the hosts through a Luke Thomas own goal and Moises Caicedo finish turned the game on its head.
Patson Daka equalised just before half time, but a second half brace from Alexis MacAllister followed Leandro Trossard's strike was enough for the Seagulls to overcome the visitors, with Leicester left bottom of the league.
Managerless Bournemouth came back from two goals down to win 3-2 in a thriller at Nottingham Forest.
Ivan Toney scored a hat-trick as Brentford beat Leeds 5-2 at the Community Stadium.
Daniel Podence scored as Wolves beat Southampton 1-0 at Molineux for their first league victory this season.
Newcastle and Crystal Palace drew 0-0 at St James' Park.
