Perez charged back from ninth on the grid with several crucial passes to join winner Max Verstappen on the rostrum, as Red Bull broke a 35-year record by scoring its 12th victory in succession.
The Mexican's race came after he'd ended a run of five grands prix which he'd started outside the top 10 on the grid, only mustering a single podium in that period at the Red Bull Ring.
The return of Daniel Ricciardo to a race seat with Red Bull's sister squad AlphaTauri has widely been interpreted as a move to evaluate the Australian for a potential return to the top team, although Red Bull has publicly stood by Perez - who is contracted to the end of 2024.
Speaking to Sky Sports after the race, Horner praised Perez for producing "a great race today".
"His recovery, his overtaking was brave," said Horner.
"His pace was fast, the way he made the places, passing cars, passing Fernando [Alonso], passing [Oscar] Piastri; the strategy that he had, he was on fire today.
"A race like that only gives you a huge amount of confidence.
"I think his overtaking, the bravery that he showed, for me that was really a statement drive today to say 'look, don't write me off'."
Horner also reflected on the team's record-breaking feat that surpassed the 11 wins notched by McLaren in 1988, having extended a winning run that began in the 2022 Abu Dhabi Grand Prix.
Red Bull remains unbeaten in 2023, with Verstappen winning nine of the 11 grands prix to date and Perez cleaning up in the other two.
Horner said: "As a young kid I remember watching the McLarens of Alain Prost and Ayrton Senna achieving that incredible feat, to think we've now bettered that...
"They were an incredible team, Ron Dennis was an incredible team principal, and to think that we've now bettered that is something the whole team here in Budapest, in Milton Keynes, everybody behind the scenes has worked so hard for and will mean so, so much."
