Playmaker Cole was a revelation in the 2023-24 season — his first at Chelsea — when scoring 22 league goals, but he has struggled with injuries and a loss of form since, culminating in Palmer missing out on a place in England's World Cup squad.
But in front of England coach Thomas Tuchel, Palmer curled home a stunning shot from the edge of the area for Chelsea's fourth goal in an end-to-end thriller.
Romeo Lavia and Pedro Neto put Chelsea 2-0 ahead by the 14th minute before Joao Pedro added a third in the 32nd for Xabi Alonso's team.
Malick Yalcouye pulled a goal back for Brighton just before halftime, and the visitors further reduced the deficit when Pedro deflected a cross from Pascal Gross into his own net in the 63rd.
Palmer's goal clinched a second straight victory for Chelsea, who opened their league campaign by beating Fulham 3-2 on Monday. Palmer scored in that game, too.
Brighton netted their third, through Gross, with the last kick of the game.
Emi Martinez played as goalkeeper for Chelsea hours after completing his move from Aston Villa.
Later in the day and trailing — again — to a promoted team, Man United were facing further embarrassment early in this season's campaign.
Then Bruno Fernandes intervened with a clinically taken hat-trick to inspire United's dramatic comeback.
United opened last weekend with a shock 2-0 loss at relegation favourites Hull and coach Michael Carrick appealed for calm.
But he saw his team fall behind on Sunday to Ipswich, another big candidate for relegation, in the 29th minute.
Then Fernandes equalised with a thunderous left-foot shot in the 40th, converted from the penalty spot for his second goal in the 61st to make it 3-1, and completed his hat trick seven minutes later by calmly slotting home a finish.
"Scoring goals is the best feeling in football," he said.
"Doing a hat trick at home with our fans is something I'm proud of. I hope I can do much more."
Between Fernandes' goals, there was time for an own-goal following a shot by Harry Maguire, who handled the ball in the build-up.
A foul wasn't awarded because it wasn't adjudged a deliberate handball by Maguire and also he didn't score directly from his shot, with the ball instead deflecting in off Ipswich defender Jacob Greaves.
Bryan Mbeumo completed the scoring for United, running onto a pass from Fernandes and lobbing the goalkeeper before Ipswich grabbed a stoppage-time consolation.
"It's just a week in football, isn't it? The ups-and-downs," Carrick said.
"We had mini-setback going a goal behind but we showed stubbornness, belief and togetherness, and that was pleasing for me."
In Sunday's other matches, Sunderland beat Fulham 1-0 thanks to substitute Wilson Isidor's goal for a first win of the season and Dominic Calvert-Lewin scored Leeds' equaliser in a 1-1 draw with Brentford.
