Polesitter Verstappen had built a comfortable lead over second-placed Alonso, but with rain looming the Aston Martin driver enjoyed a strategic advantage by starting on the hard tyre, while Verstappen began to struggle on worn mediums.
As the rain eventually hit and then intensified, Verstappen pitted on lap 56 of 78 for intermediates and managed to undo Alonso's tyre advantage as well as the Spaniard's bid to benefit from any safety car or red flag.
According to Horner, Verstappen's pitstop came one lap too late, but Alonso's gamble to bolt on a new set of mediums instead of intermediates immediately backfired and relieved the pressure on the Dutchman as he survived a scare at Portier to take the win.
"I think we were on the ropes here," Horner told Sky F1.
"We knew coming into this weekend, this was going to be our biggest challenge in the first half of the year and the low-speed nature was playing to their strengths. We actually thought Ferrari might be the main opponent, but Fernando's just been on fire all weekend. I mean, unbelievable.
"We were nervous from the moment the red light came on, because when they pulled the covers off, you could see quite a few of them went on the hard tyre and there's rain around, if they can go far enough, if the mediums degrade...
"It felt like maybe we had gone a lap too long before going on to the intermediates. I think had Fernando picked the inters, it would have been much tighter and put more pressure on our pitstop.
"But thankfully, he picked the slick tyre just as the rain was going. At that point, it's like: 'Max don't take any risks, just get it to the pits.
He added: “I was surprised they took the medium tyre, so that totally let us off the hook."
Horner praised Verstappen's mutual understanding with his long-time race engineer Gianpiero Lambiase.
"There's a lot of trust between the two of them and that relationship between engineer and driver is so unique," he explained.
"There has to be a bond between the two that they can just speak openly and honestly and you're managing the driver's emotions, giving him the the information and there's got to be that trust between them which you hear time and time again with those two.
"When you're the lead car, you can only lose, you've got nothing to gain. And you've just got to try and work with the conditions."
