Coach Maurizio Sarri was sacked the following day, with Pirlo - only recently named their Under-23s boss - promoted as his replacement on a two-year deal.
The World Cup winner took to social media on Sunday and said: "I'm deeply pleased and honored to receive such respect and trust from Juventus.
"Ready for this amazing opportunity!"
Conte welcomed the news as he faced the media ahead of Inter's Europa League quarter-final against Bayer Leverkusen.
"I'm certainly very happy for Andrea," said Conte, whose Nerazzurri finished a point behind Juve in Serie A.
"Besides what he gave me at a football level, we are talking about an exceptional person.
"What I think if there are players who I previously coached and are now coaching is that it means that I'm getting old!"