The tyre delta was created by Red Bull opting to pit Verstappen early during both pit windows after Hamilton initially got the jump on the Dutchman off the front row.
Hamilton cut a gap of over eight seconds down to just one by the chequered flag with Verstappen nursing eight-lap older tyres and Horner told Sky Sports F1: "I really didn't think he was going to pull that off.
"Lewis, eight-lap newer tyres, the first set of hards that had come off, we were pretty much down to the canvas and then thinking 'we aren't going to have much left at the end here'.
"Really, the way he managed that tyre through the last stint to make sure he had enough for the last five or six laps, we talked about getting out of turn 11 at the beginning of the race, it was about getting out of it at the end that was going to be crucial."
Horner has experience of losing a lead at the end of a race after a long chase after Jenson Button got the better of Sebastian Vettel in the 2011 Canadian Grand Prix.
"Lewis, you know at the end of the race he is so strong and he went long and bought himself an advantage," added Horner.
"To lose a race in the last two or three laps... we lost one in Montreal  and that really hurt. To lose one here would have been really painful.
"Max just held on, he did a great job and managed the race really smartly and had just enough at the end there."
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