Chelsea made the fastest possible start, with Joao Pedro finishing off a Cole Palmer through ball inside 35 seconds. Josh King equalised for Fulham in the 23rd minute before Morgan Rogers restored Chelsea's lead on his debut and Palmer added a third early in the second half.
Gonzalo Garcia pulled one back late on, but Alonso's side held on for a winning start under their new head coach.
Speaking to Sky Sports afterwards, Alonso reflected on the balance within his squad.
"I think overall the squad is balanced and we need to have good foundations from the back and middle to connect with these players, with the special players. They need to have the freedom but as well be responsible to help the team and accept that responsibility and once they get that flow, they connect well," he said.
"We know that they can create many chances, they have this talent, but not just talent they have teamwork so yeah, they have a good start but we can't rely on them.
"The whole team needs to have stability, have control, quality and sharpness to reach them in the best possible positions."
Asked specifically about Robert Sánchez's afternoon, Alonso was measured.
"This is all about the team and you know it's not about individual situations — it's about understanding what it takes for sure. The first goal we can do better than that but it's a process. It's day one and normally it's easier to learn with three points."
On the ongoing transfer window, he added: "I would love for it to be already closed but it is what it is and we need to be ready for any situation. You never know with football so let's see."
Speaking separately to BBC's Match of the Day, Alonso expanded on his satisfaction with the win.
"I think it was a professional performance. The game had different momentums. To win away in the Premier League is normally tough. We were prepared for that. Happy with the performance. It is a good first day," he said. "Great combination, great quality. We know if we reach them [Joao Pedro, Morgan Rogers and Cole Palmer] in good positions, they have this talent to create special things and I am very happy the three forwards have scored."
On Palmer's display in particular, Alonso was effusive.
"I have really enjoyed his performance. He has played with responsibility, with maturity but as well with freedom. It is very important for me that we get the best from Cole when he takes this role. He feels where he has to be involved and I am happy for him. If we want to have a good team and we need a great Cole."
He tempered the optimism, though: "It is still a process for sure. We need to keep working, analysing and improving. Every game is a lesson." Asked about Enzo Fernández's future, Alonso said: "In football, I am not surprised by anything. We will see. My wish is that we know what will happen for the future but we will see this week."
For Fulham, manager Álvaro Arbeloa was left to reflect on a difficult opening afternoon in his own first league game in charge.
"We concede goals in the first minute, the end of the first half and the beginning of the second half and there are not worse moments to concede. But we showed character and personality and I am very proud of how my players responded. We deserved a little bit more but football is not about what you deserve, it is about what you take," he told Match of the Day.
"If you lose against a team like Chelsea they [should] have to do a little bit more. For me, they are easy goals for them and mistakes we cannot make. We need to improve and be more ready in the first minute, that is for sure."
Arbeloa found some encouragement in his side's response.
"We are on our way and we will keep improving. The first thing I want to see is a competitive team. We were tonight, we tried our best and tried to play our football against a team like Chelsea, which is not easy."
He also praised his younger players.
"I believe in them and I believe in young players. I think they have the energy and ambition I want and they fit in the way I play. Josh King is absolutely amazing. What a player you have here and I am really happy with him. Gonzalo scored and Cesar did well. I think they need a bit more time but they are doing great and making an impact already."