The fire forced the Tennessee rockers to cancel Tuesday night's concert - the finale to their European tour.
London Fire Brigade said 12 fire engines were called out to battle the early morning fire in a loading area. Six people were treated for smoke inhalation, but no one was hospitalized.
Seating areas of the arena were unaffected, but the band said in a statement that "the complications and disruption caused by this morning's fire have meant there is now insufficient time to rig the arena for tonight's performance."
The show will be rescheduled for next year.