A rare full-length portrait by Austrian artist Gustav Klimt has been sold for £24.8m (almost $40m), with fees, at an auction at Sotheby's London, passing the estimate of £12m-£18m. Bidding starting at £9.5m.

Klimt's 1902 portrait of Gertrud Loew, the 19-year-old daughter of Dr Anton Loew, the director of a prominent Viennese private hospital, had never previously been offered on the open market.

It was purchased by an unknown buyer over a mobile phone, while another prospective buyer tried to outbid the winner via a telephone connection with a Sotheby's staff member. "Is everyone awake still?" quipped the auctioneer Henry Wyndham, chairman of Sotheby's Europe, to the audience, halfway through the duel that lasted more than a quarter of an hour.

The Guardian reported that it took 15 minutes to get from the opening bid to the hammer bid of £22m. "I'm sorry about this," joked auctioneer Henry Wyndham as the bids slowly went up by £100,000. "I'll be with you later."

At £19.1m, Wyndham observed: "Have we all had enough of this?" When it finally sold there was applause.

"A painting by Klimt on this scale is very rare," London-based art consultant Patrick Legant, a former Sotheby's specialist in Impressionist and modern art, told the New York Times. "It's a jewel. It really gives you a feel of the Vienna of that time."

Dr Loew sanatorium had treated turn-of-the-century Viennese VIPs such as Mahler, Ludwig Wittgenstein and Klimt himself.

The portrait of Gertrud Loew was the first major portrait by Klimt to sell at auction since the 1917-1918 Portrait of Ria Munk III, which raised £18.8m, at Christie's in 2010.