The teenager, who was only eight years old when Rosberg made his Formula One debut in 2006, finished fourth at the Circuit Gilles Villeneuve with the vastly more experienced Mercedes driver fifth.
Rosberg tried repeatedly to pass in the closing laps but Verstappen, who last month became Formula One's youngest winner with victory on his team debut in Spain, showed plenty of race craft and determination to keep the German at bay.
When the Mercedes driver did get past, he spun and lost the place again.
"He’s shown great mental strength," said Horner of a youngster who followed up his Barcelona win by crashing in Monaco during qualifying and the race.
"Monaco was a tough race for him but the way he has bounced back here, that confidence that he’s had, throwing the car around, it hasn’t affected him at all.
"It was a pretty mature drive for somebody that Mr Wolff said was not quite as mature as he thought. He looked pretty mature to me when he was fending off his very experienced mature driver," said Horner.
Wolff had told Canadian reporters earlier in the week that he felt 17-year-old Canadian Formula Three driver Lance Stroll was more mature than Verstappen.
"When you speak to the boy who's only 17 years old, you tend to believe you're talking to a 35-year-old," he said. "His maturity and his intelligence are going to take him far. I'm sure this kid will be the next Canadian (driver) in Formula One."
Comparing Stroll to Verstappen, the Austrian said the Dutchman acted more like a teenager.
"In the car he's extremely good but when you interact with him, he's a kid. But with Lance, his maturity is very surprising," he said.
(Writing by Alan Baldwin in London, editing by Pritha Sarkar)
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