Poleman Verstappen was overtaken by George Russell at the initial start, and he then lost a further spot at Turn 3 when, in Horner's words, Hamilton "sent a missile" down the inside and made an aggressive pass.
Verstappen then sat in third watching the Mercedes drivers battle before Russell pitted when a safety car came out after Alex Albon crashed. The subsequent red flag gave Hamilton and Verstappen a free tyre change and kept them at the front of the order.
A few laps after the restart, the Red Bull driver took the lead with a DRS move and began to pull clear on his way to an eventual victory.
"Obviously Max had to make the pass on Lewis," said Horner. "The two Mercedes were fast starting, they were very aggressive on the first lap, Max showing great patience really not to get embroiled.
"George got a blinder, and he just sent it down into the first corner so Max got out of that, and by getting out of it to allow him through that then compromised him through T2.
"And then Fernando [Alonso] and Lewis get a run on him and then Lewis sends a missile down into T3."
Horner praised his driver for leaving space for Hamilton.
"Again, I thought he showed great patience to get out of the way," he said. "It was hard racing between the two of them, and then of course the race was just getting going, and it was quite a fight between the three of them at the front.
"Sounded like George wasn't happy about being pushed by Lewis, who in turn was being pushed hard by Max. Then they started having a go at each other, then they pitted George, then the red flag came out and then it was just a straight fight between Max and Lewis, and he made very short work of that.
"Then it was a question of controlling the race from that point onwards."
After the race, Verstappen made it clear that he wasn't happy with Hamilton's first lap pass.
"From my side, I just tried to avoid a contact," said the Dutchman. "It's quite clear in the rules what you're allowed to do now on the outside. But clearly it's not followed.
"But that's OK, we had good pace and we passed them anyway. But it's something for the next races to take into account."
Hamilton in contrast believed that his move was a fair one, with both men giving each other breathing space.
"I thought it was pretty decent," said the seven-times world champion. "I mean, he braked early and I braked late, and I was fully up the inside, and I think we both left space for each other.
"I didn't run him off the road and he didn't turn in on me. So we didn't touch, and that's racing."
Mercedes boss Toto Wolff also saw no issue with Hamilton's move.
"We have lots of discussion in the past about racing rules," said the Austrian. "I don't think that was a bad one."
