Taken from The Phrase Finder:
This is early aviation parlance. Aircraft initially had few navigation aids and flying was accomplished by means of the pilot's judgment. The term emerged in the 1930s and was first widely used in reports of Douglas Corrigan's flight from the USA to Ireland in 1938.
That flight was reported in many US newspapers of the day, including this piece, titled 'Corrigan Flies By The Seat Of His Pants', in The Edwardsville Intelligencer, 19th July 1938:
"Douglas Corrigan was described as an aviator 'who flies by the seat of his pants' today by a mechanic who helped him rejuvinate the plane which airport men have now nicknamed the 'Spirit of $69.90'. The old flying expression of 'flies by the seat of his trousers' was explained by Larry Conner, means going aloft without instruments, radio or other such luxuries."
The 'old flying expression' quoted above (although it can't have been very old in 1938) that refers to trousers rather than pants does suggest that the phrase was originally British and crossed the Atlantic prior to becoming 'by the seat of one's pants'.