Haaland headed Manchester City in front in the 17th minute, meeting Antoine Semenyo's cross from close range, before Cherki doubled the lead with a driving solo run and finish just after the hour.
Gianluigi Donnarumma turned a Yeremy Pino free-kick into his own net moments later to give Palace hope, but Cherki restored the two-goal cushion within two minutes and Haaland completed the scoring late on to make it 4-1.
Crystal Palace remain winless under new boss Pierre Sage. Speaking to Sky Sports, he said: "I think we were able to play 60 minutes, stayed in the game. When I made the substitutions, the energy of the team went down. I'm a bit angry with that."
Asked whether that frustration was directed at himself or his players, Sage added: "Both, we are together. I'm with my team every moment. We had a lot of chances, it's difficult to have hope against this kind of team without scoring. When you change players, you need more energy, I don't find that.
"We know we are in a process, we know we have to work. We have to not accept this kind of situation, we deserved a better result. Two negative situations at the end. We were late in our preseasons. I think at the moment we need substitutions. You can't replace fatigue with low energy."
For Maresca, it was back-to-back wins to open his City reign.
"It is a tough pitch, we knew that before the game," he said. "The intensity on the team has been good. We scored four, probably could've scored more, only conceded one. Off the ball we were quite good, struggled for the first 10 minutes. On the ball, tried to put players into a position to attack."
On Cherki: "He's helping us and is going to help in the future. We know he's a quality player. I think he was very, very good off the ball and was working very hard." On Foden, who set up two of City's goals: "He's adapting, we know he's not a winger, we're trying to find a solution. When he was inside, you could see he was enjoying it more. The attitude is very good and it's what we need."
Maresca added: "It's important to start the right way with a new manager. Big change. You can see at the end of the game how happy the fans were. Happy that we get the three points. There are always things that we can do better."
Elliot Anderson, starting his third game since his summer move from Nottingham Forest, said he was settling in quickly: "I felt more at home today, it's my third game now and I feel like I'm getting better. There's still more to come."
On his nerves before his first Etihad appearance, he added: "Yeah, of course. I'm always nervous when I go on the pitch but my first outing at the Etihad was pretty special. But it's two wins from two now so I'm pretty happy. I'm loving it. It's a bit different to what I've been used to. I've got to be a bit more structured, that's what I'm learning - to stay in my position when at Forest I was doing a little bit more."
On Cherki's second goal, Anderson said: "I'm just sat back watching. I see this every day since I've joined, him, Phil... they've got so much talent. It's no surprise, to be honest. The talent they have is scary."