The Dutch Grand Prix was derailed by two rain showers, the latter of which prompted a red flag due to severe aquaplaning into Turn 1, which caused a crash for Alfa Romeo's Zhou Guanyu.
At the start of the lap 62 downpour Verstappen wasn't entirely convinced about coming in for intermediates yet, with his race engineer Gianpiero Lambiase urging him that "it would be a good time to pit" as the track conditions worsened rapidly.
Afterwards team principal Horner revealed the call to err on the safe side and bring in his driver came from him.
"We were a bit unsure of what was going to happen with the weather; was it a few spots, was it going to blow through?, Horner said.
"And in the end, I was shouting at GP: 'Get him in!'"
Horner says his team nailing the strategy, on a day when the likes of Mercedes, McLaren and Williams didn't, was part luck and part risk management.
"I think it's a combination of the two," he added. "There were so many things that could go wrong.
"We saw calls being made early on where cars dropped out of contention, into contention.
"But we've done eight pitstops today and the team has nailed every one of them other than the one that Checo [Perez] gave us about five seconds notice for. So, it's been an incredible team performance."
The first rain shower occurred during the start, with seventh-starting Perez immediately opting to pit while Verstappen stayed out longer and lost the lead to his team-mate.
"The first car in the train was the leader, so it's very hard to make that that call when it's right on the edge," Horner said about Verstappen's initial reluctance to pit after lap one.
"It was Sergio's call that that he wanted to pit and that paid dividends for him, and he was the first one certainly from the front-running cars to jump onto the inters. Max came in at the end of the next lap.
"There was debate whether the rain was going to stay or go or not, but he went straight in and then the way he cut through the traffic was quite sensational."
The roles were reversed when Verstappen undercut Perez on the change back to slicks 10 laps later, a decision which puzzled the Mexican at the time.
Horner said the call was made because while Perez was out in front, Verstappen was vulnerable to an undercut attempt by Aston Martin's Fernando Alonso and others.
"Checo had the target of pulling a gap on the cars behind him, but Max's pace as he came through the field was such that I think at one point he took seven seconds out of him in three laps and he was coming very, very fast," Horner explained.
"And then you get to that awkward point of the crossover and Alonso and I believe it was [Pierre] Gasly behind him [it was actually Zhou, ed.] had pulled the trigger and pitted.
"You could see from Albon suddenly the circuit was coming alive for the slicks. Had we pitted Checo first, they would have both undercut Max, so we'd have gone from a net one-two to a net one-four.
"So, we pitted Max first, with the risk being that he'd undercut Checo, but we'd end up with a first and second as a team, so it was a no-brainer."
Additional reporting by Erwin Jaeggi
