Booking began this morning at 7.30am, but was halted after the site crashed due to popular demand.
Disappointed fans then tried to book via phone, only to discover the phone lines were also jammed.
Later in the morning a message appeared of the site: “Kraftwerk at Tate Modern has just gone on sale and we are experiencing a phenomenal demand for tickets which is affecting our web server.
“We have extra staff on hand today but demand is extremely high.”
The Tate’s director Nicolas Serota defended their website by comparing it to the London 2012 site.
“This isn’t the first website to crash through over-demand,” he said to the BBC. “The Olympic site did something similar.”
“There are a limited number of tickets, there is a huge demand and I hope we can satisfy a good many of them.”
Many Twitter users were far from impressed: “If you can’t cope with demand get some professional ticket sellers who can,” wrote user Pete Gleadall.
The upcoming Tate shows, will be the first time the band has played in London since 2004.
A message on the Tate’s website reads ‘We are currently experiencing an extremely high demand online for Kraftwerk tickets. Please call the ticketing line on +44 (0)20 7887 4919. There is a capacity of how many calls can come through so please keep trying.’