Responding to questions following another day of claims and counter-claims for the troubled champions, Horner said everyone in the organisation, made up of 1,400 people, played a role in their successes.
But he avoided giving full comments on the latest spats which had suggested senior adviser Helmut Marko faced possible suspension before a pre-race meeting at the circuit after which he made clear he was staying.
Three-time world champion Verstappen said on Friday that he would consider leaving Red Bull if they parted company with his long-time mentor Marko.
"Obviously a lot is made of this stuff," said Horner when it was put to him that there was an internal power struggle for control of the team.
"We are one team and nobody is bigger than the team. This team comprises across different entities over 1,400 people.
"Everyone has a role to play and that is from the very bottom to the very top. Without them performing, you don't achieve performances like this.
"Unfortunately there has been a lot of speculation this weekend but our focus has been very much on the track."
Marko's future was the subject of speculation on Friday following reports that suggested the 80-year-old Austrian may have been the source of leaks to news media over complaints by a female team member about the conduct of Horner.
On Marko's role and influence, Horner said: "Helmut Marko is a consultant to Red Bull GmbH and not the team. Max is a valuable member of this team and a wonderful driver. We are a team - and no single individual is bigger than the team."
Clearly seeking to regain some control of the narrative after weeks of negative publicity - often focussed on allegations he subjected a female collegaue to "inappropriate behaviour" - Horner said he was aware of the rumoured divisions.
'Aware of rumblings'
"I am aware of rumblings and I am aware of what has been said," he said when asked about Verstappen.
"He is our driver and he's doing a great job and as team principal and CEO I'm responsible for the running and the operation of this team. Everybody has to do their part."
He evaded again when asked by Sky Sports F1 if he needed to make an effort to smooth things over with Verstappen's father Jos, who last weekend declared that Red Bull would explode if Horner remained.
"I think that speculation is obviously rife and the most important thing is the team and it has been a phenomenal team effort. You only see the front end and I think it's important to recognise it's not that easy.
"You only do that by having a spirit, a culture and determination around the company.
"Red Bull Racing is the most followed sporting entity of any sports team in the world. We have an incredible team of men and women without which we wouldn't be achieving these performances."
The female team-mate who claimed she had been subjected to Horner's "inappropriate behaviour" was suspended on Thursday.
Horner, 50, has strenuously denied the claims and was cleared by Red Bull following an internal investigation.
Verstappen opted to keep his comments focused on racing rather than his future with the team.
"Overall, it's been a fantastic weekend for the whole team and also for myself and for the car. I felt really good," said the Dutchman.
He had confirmed before the race that he was satisfied to learn that the 80-year-old Marko was not facing suspension or expulsion.
Horner and others stuck to the "team unity" script.
He added that the race had been physically tough - a high-speed street circuit, anti-clockwise and in searing heat.
"It's early in the season, it's anti-clockwise with a lot of G-force to hold up - it is one of the more physical ones," he said.
