Having made a brilliant initial launch from fourth on the grid, Perez was able to draw alongside the two Ferraris ahead of him and make a lunge to grab second place behind polesitter Verstappen.
But the Mexican was unable to get his car slowed down on time as he hit the brakes approaching Turn 1, nearly hitting the back of Verstappen's similar RB20 as they navigated the tight right-hander.
Perez eventually ran wide over the tarmac run-off area and rejoined the track in fifth place, behind not only Ferrari duo Charles Leclerc and Carlos Sainz but also McLaren's Oscar Piastri.
Horner feels a poor getaway for second-placed Leclerc opened up an opportunity for Perez at the start, but Red Bull came close to having both its cars involved in a painful first-lap incident.
"His start was optimistic. Charles didn't have a great start ahead, I think that caused Checo... he had to lift for Charles and then he had a window into the first corner," explained Horner.
"He went for it, obviously got in very deep and was lucky not to collect Max at the first turn and not to collect the Ferrari coming back onto the track.
"So I was pleased to see most cars survive that."
Verstappen revealed that he saw Perez closing in on him at Turn 1, but a disaster was averted as the Mexican just about managed to avoid crashing into the triple champion's car before running deep into the corner.
"I was very aware. I turned in and I saw him lock up," he said. "And I looked after the race, there was like a scratch on my diffuser. So something must have hit.
"But yeah, it was very close. Could have ended in a disaster, of course, for the team as well. So yeah, lucky."
Lacking the pace to mount a significant recovery, Perez could muster just fifth at the flag, which became fourth when Sainz was hit with a post-race penalty for an incident with Piastri after the safety car restart.
Perez admitted that he had to take avoiding action after coming perilously close with Verstappen in a corner that had already witnessed a race-ending clash between McLaren's Lando Norris and the Aston Martins in Saturday's sprint.
"I had a good start, Charles had a really bad start, but as soon as I brake into the inside there was no grip, like with Lewis [Hamilton] yesterday [in the Sprint]," he explained.
"And offline there was no grip, and I ended up locking. I nearly took off Max out. So I had to come off the brake and I lost a position to Oscar.
"It was quite unfortunate, but other than that, I think, we were struggling for some pace today. We just couldn't manage to get a pace where we needed and, yeah, something that we got to work on to try and understand why, what was the issue."
Additional reporting by Filip Cleeren and Ronald Vording
