The play, written by Jack Thorne, is set 19 years after the seventh and final book in the series by JK Rowling.
It opens officially at the Palace Theatre, in London's West End, on Saturday.
Audiences have been urged to "keep the secrets" since the play began previews in early June.
Presented in two parts, the play - showing the stars of the wizarding saga as adults in their mid-30s as their own children head off to school - stretches over five hours.
Daily Telegraph critic Dominic Cavendish awarded the play five stars, writing: "British theatre hasn't known anything like it for decades and I haven't seen anything directly comparable in all my reviewing days."
He said "those involved can give themselves a pat on the back", adding: "It's a triumph. Not an unqualified one - there are some quibbles - but in all key respects, it grips, it stirs, it delights."
Cavendish praises the "thrill-a-minute" stage craft which sees pupils heading to Hogwarts, at the start of the play, change into school uniforms "in the blink of an eye" as they head to the infamous Platform 9Â¾ to make their way to school.
The Stage also gives a five-star review, with Mark Shenton describing Harry Potter and the Cursed Child - co-devised by Rowling, with director John Tiffany and playwright Jack Thorne - as a "truly game-changing production and a thrilling theatrical endeavour".
It is an "entirely original" piece of work, with "Dickensian sweep and momentum to the storytelling", writes Shenton, adding: "It earns its place on the stage, feeling distinct from both the books and the screen adaptations.
"By turns playful and gripping, disturbing and detailed, poignant and powerful, it is superb family entertainment."
Shenton also praises "the theatrical wizards who've created this stunningly-realised alternative universe deliver one coup de theatre after another".
In a nod to the Kings Cross platform number, Ann Treneman gives the play four and three-quarters stars in The Times, saying the "wizardry on display" is "out of this world".
She says the plot is complex ("there are mazes that are more straightforward") but adds: "It's a raunchless Game of Thrones with heart. Crucially, it's authentic Potter but, most importantly, it's new.
"It's not the movie of the book. It's the real deal, live in front of you, so much better than any film could be."
In The Guardian, Michael Billington noted the Cursed Child will make more sense to "hardened Potterheads", but applauded Tiffany for directing a "thrilling theatrical spectacle", giving it four stars.
He praised the strong performances that meant acting was central to the story, despite the dazzling special effects, singling out Sam Clemmett as Harry's son Albus, "wonderfully quirky" Anthony Boyle as his friend Scorpius Malfoy, and the adult Harry (Jamie Parker), "authoritative" Hermione (Noma Dumezweni) and "bluntly commonsensical" Ron (Paul Thornley).
It's another five stars from The Independent, with Jack Shepherd describing it as "magical". He argues Part One should be billed as a magic show, due to the effects used, also praising its moments of comedy.
Shepherd adds: "It's quite apparent this isn't written to be either a book or a tie-in film; it's a spectacle for the theatre, one that is filled to the brim with fan service and magical imagery that will amaze."
Quentin Letts grumbles about the length of the play in the Daily Mail, noting: "There were moments I could have done with a glug of gurdyroot infusion to keep me alert."
But he admits: "Potter addicts will love it. JK Rowling is going to make (another) fortune. The West End's ornate Palace Theatre, itself a little like Hogwarts, has a hit probably for years."
Variety describes it as "spellbinding", Matt Trueman writing that it is: "The Show That Lived Up to Expectations â€” and Then Some."
He says the relationship between Albus and Scorpius is "the friendship of two bullied boys bound together, and it's a beautiful, tender thing", adding: "The script by Jack Thorne recognizes that rejection breeds resentment, and outsiders stew into outcasts.
"No one's born a villain, nor sees themselves as such, and where the books gave us stock baddies, Cursed Child fleshes them out."
Chris Jones, in the Chicago Tribune, says that "heretical as this may sound", the play left him "quietly lamenting that the movies were ever made".
Ben Brantley in the New York Times writes: "Like the novels that preceded it, The Cursed Child is stuffed with arcana-filled plots that defy diagrams and baldly wrought sentimental life lessons, along with anguished dives into the earnest, tortured solipsism of adolescence.
"By rights, such a combination should try the patience of any grown-up. But like Ms Rowling's books, the play vanquishes resistance."
The Harry Potter books have sold more than 450 million copies since 1997 and been adapted into eight films.
The script of Harry Potter and the Cursed Child is published this weekend.
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Harry Potter and the Cursed Child has won five-star reviews from critics, with one describing it as "a game-changing production".