Seidl's move to Sauber was announced late last year, and Stella was immediately named as his successor as team principal.
Stella joined McLaren from Ferrari in 2015 and has had various job titles, latterly serving as racing director.
Brown says that Stella's knowledge of the team made the transition much easier than if an outsider had come in and had to learn his way around the team.
"It's been very seamless," said the American. "The change almost effectively happened overnight. Andrea settled in the next morning.
"Obviously he is very aware of the team, he's worked very closely with all of them, he's been here for almost a decade. He's exactly who we wanted to have run the racing team.
"Andrea slept on it a little bit, he hit the ground running and it's been a continuation from our point of thought."
Stella agreed that his experience with McLaren made it easier for him to take over the new role while conceding that he had to learn more about some areas of the team.
"The points of strength of this transition were two, I think," said Stella. "The first one is the knowledge of the team on my side, and then the second one is that some important projects had already been started before.
"Saying that, you are obviously stepping into the position of team principal. Compared to what I was doing before, I have to learn a lot more, like much more into the technical side or the manufacturing side or in general getting all these areas to work as efficiently as possible, and totally focused on performance.
"So how do you achieve it? You achieve it by having many, many conversations. You need to know the place inside out, assess this together with your people, get the people on the same page, understand the mission.
"Actually, firstly, state the mission, get everyone to understand it, and then start to work on implementing the plan.
"So elements of continuity, which were good to start from, but then obviously we will move into a lot of details as to when it has to be implemented, and so on."
Lando Norris paid tribute to Seidl but stressed that he has complete faith in Stella's ability to fulfil the team principal role.
"I think Andreas did a very good job with what you see now, with what you see McLaren being currently, and the change from the past few years," said Norris.
"The structure we also have within MTC, the atmosphere, Andreas did play a very good role and has been a team manager of many different teams, and very successful teams. So he did play a big part, and I think that's big shoes for Andrea to fill.
"But at the same time I have full trust in what he's capable of doing. And he can also bring a different perspective. He's a guy who was a race engineer, so he also sees and understands things very much from a racing point of view. As did Andreas in many ways.
"He's not been in that position long, but he's done very good job with boosting the team, making sure that morale is up, especially so early on in the season, giving a very clear and concise plan of combating our issues, our weaknesses, and improving in all these areas.
"And from the feedback I've got from people around the factory, I think everything is settled in, he's settled in very quickly, and people are very happy with the job and the feedback that he's given. So no lacking in confidence or whatever. I'm very happy with how things are."
