Fans leapt to their feet for a five-minute standing ovation at the Palace Theatre in London on Saturday night.
Tiffany, who thanked the actors and the crew, joked that he was planning to get very drunk in celebration of a successful gala opening.
The all-day event saw part one of the production staged in the afternoon, followed by part two in the evening, completing the story.
Pottermania continued when the play script was released at midnight, enabling fans across the globe to find out what happens next to Harry Potter and his friends.
Speaking on the red carpet at the gala opening, Rowling revealed that fans unable to make it to the West End production could soon also have a chance to see the story on stage - as the play could be destined for Broadway and beyond.
Asked about Broadway plans, she told reporters: "I'd love it to go wider than that. I'd like as many Potter fans to see it as possible."
Theatre producer Sonia Friedman said "many countries" could get a chance to see the play in future years.
She said: "Hopefully more than America, hopefully many countries at some point will get to see it. But it's a big piece of theatre, it's a big endeavour, you can't just turn it around overnight. But if everything goes to plan over the years, we will get there."
As the play opened following nearly eight weeks of previews, it drew whoops, applause and gasps of shock from the audience as magic appeared to unfold on-stage.
The play also features plenty of twists and surprises, although fans have been asked to keep plot details secret – with #KeepTheSecrets badges handed to audience members on their way out.
Rowling said she had been impressed that fans had kept details under wraps.
She said: "It is the most extraordinary fandom so I'm kind of not surprised they didn't want to spoil it for each other but I'm so happy we got here without ruining it."
The Harry Potter author highlighted the importance of the Friday 40, a chance for people without tickets to win seats at low prices. "What we would really like most of all is to bring people in who have never been to the theatre before," she said.
"I would be so proud to think that kids from my kind of background, who didn't come from particularly theatre- going families, learn what theatre is about through this show. That would be an incredible thing."
Attending the gala with his family, Mayor of London Sadiq Khan spoke of his pride that the play was launching in the capital.
He said: "Many thanks to JK Rowling for ensuring the premiere is here", and added that he was a "big fan" of Harry Potter, saying: "What's important is that the world premiere is here in London, and we should be really proud."
Set 19 years after the events of the seventh and final book, The Cursed Child brings back Harry Potter, now grown up and an employee at the ministry of magic.
Harry and his wife Ginny Weasley wave their youngest son Albus Severus off to their old wizarding school, Hogwarts – but once there, Albus struggles with the weight of his family legacy and goes to extreme and dangerous lengths to right the wrongs of the past.
Reviews have been glowing, with Daily Telegraph critic Dominic Cavendish saying that "British theatre hasn't known anything like it for decades".
The two-part play stretches over five hours and was co-devised by Rowling, written by Thorne and directed by Tiffany.