O2 customers were the first to have access at 10:00 GMT, but some reported problems with its app and website.
Those who did manage to purchase tickets reported prices ranging from £56 to £199, with VIP "on stage" seats at a wallet-busting £1,950 to £2,390.
There will be further pre-sales on Friday and Monday.
Tickets for the pop superstar's five UK stadium shows in Cardiff, Edinburgh, Sunderland and London will then go on general sale on Tuesday.
The UK concerts are part of a 43-date world tour in support of her Grammy-nominated Renaissance album.