The highly-rated Piastri has signed to race for McLaren from next season after a contract tussle with Renault-owned Alpine.
The 2020 Formula Three and 2021 Formula Two champion raced for the Arden team, founded by Horner and his father, in Formula Four and Formula Renault.
"There was an opportunity for Red Bull to look at him at the time and we didn’t take up that option, which is something that I regret," Horner told Formula One's Beyond the Grid podcast released on Wednesday.
Piastri will have Britain's Lando Norris as his McLaren team mate next year, another driver Horner said he rated very highly.
"The only problem he (Piastri) now has to deal with is (that) the expectation on him will be immense," said Horner. "He’s going to have to get in and deliver against Lando, which is no mean feat.
"But you either sink or swim in this business. He’s a very, very capable driver, I’m sure he’s going to do very well."
Red Bull have won 12 of 16 races this season but Horner doubted his team were starting a new era of domination like the one they enjoyed from 2010-13.
Red Bull are set to take both championships with Max Verstappen heading for his second successive drivers' crown -- possibly as soon as Singapore next week.
"If we manage to pull off these back-to-back championships, which hopefully we can do with Max, and if we are able to for the first time in eight years be the only team to beat Mercedes to the constructors' championship, that's an enormous feat," said Horner.
"Now, the quality of our competitors is such that I cannot see a period of domination like we've just endured... I'm sure the others are going to come back fighting very, very hard next year but only time will tell."
Red Bull are 139 points clear of Ferrari in the standings with six races remaining while Verstappen is 116 ahead of Ferrari's Charles Leclerc.
