The Pixies, Brixton Academy, London

For acts of sufficient vintage and appropriate career trajectory, performing entire albums live has become something of a statement. It's hard to say exactly when the trend started but Lou Reed's concert version of Berlin was an important moment, as was Brian Wilson's Smile show.

Of course it helps if the album contains enough great music for the audience not to feel they are missing out on other material, which is why the format worked so well when ABC gave the live treatment to The Lexicon of Love, and Echo and the Bunnymen did the same with Ocean Rain, both of them works on which their creators were at the height of their powers and which have really stood the test of time. Those criteria are essential.

But surely no band has pulled off a one-album show as triumphantly as the Pixies did here with a 20th-anniversary performance of their 1989 masterpiece Doolittle, a collection of songs of such exhilarating decadence, crazed urgency and melodic inspiration as to earn the Boston four-piece an imperishable place in the rock pantheon.

The third of their five albums, Doolittle was sensational from the word go, a flawless distillation of the Pixies' irresistible brand of proto-grunge, punk, pyschobilly and surf music. But as with almost any supreme feat of artistic achievement it was a while before its full significance became apparent. Kurt Cobain famously said that "Smells Like Teen Spirit" was an attempt to write a Pixies song, and in containing so much of what went before them and so much of what came after, the Pixies could arguably be described as the most pivotal band in rock history.

Yet even as Doolittle crept up the all-time greatest albums lists, the feeling persisted that the Pixies were not being given their due. That may have had something to do with the particular affection in which they were held by their fans, who maybe didn't even want to share them with too many other people.

After a long and less than amicable period of break-up in the 1990s, during which charismatic bandleader Black Francis established himself as the solo artist Frank Black and bassist Kim Deal was busy leading the Breeders, the band she had formed in 1988, the Pixies amazed people when they got back together to tour again in 2004. Those shows were a knock-out, and the clamour to see them this time round was such that they sold out four nights at the Brixton Academy in 20 minutes.

The back-projection of clips from avant-garde black-and-white movies from the 1920s emphasised the intellectual credentials of a band whose surreal lyrics are a big part of their appeal. "Debaser", the opening track on Doolittle, directly references Luis Bunuel’s Un Chien Andalou, and its notorious eyeball-slitting scene added an appropriate frisson of danger to the run-up to the band's appearance.

A handful of Doolittle-related B-sides formed a prologue to the main event, and it needed Doolittle to get under way before momentum was fully established. If Joey Santiago's lead guitar lines didn't glitter quite like on the record, the tenderness of Deal's vocals provided a heart-rending counterpoint to Francis's yowlings, and the power and precision of David Lovering's drumming added up to an entire show-within-a-show.

"Wave of Mutilation" had an exquisite delicacy; "This Monkey's Gone to Heaven" soared and swooped, "Mr Grieves" encapsulated a uniquely Pixiean sense of joyous abandon. Everything was a highlight but if one could choose the moment to re-live it would have to be when the back-projection comprised the band-members’ faces as they sang along to the infectious "Here Comes Your Man". A gloriously uplifting sequence in a show that was a lesson in how a legendary band can reunite and actually enhance its status and its dignity.