The 31-year-old American’s 3-6, 6-3, 7-5 victory over Petra Kvitova in the quarter-finals put her back on top of the world after an interval of two and a half years during which she sometimes thought she might never play again.
Williams had been 1-4 down in the final set of an outstanding match against the Czech, and her relieved smile, heavenward gesture and tears in her moment of triumph gave lie to her earlier statement that the No 1 position no longer mattered.
“I am so sensitive nowadays. I am always crying, but I never thought I would be here again you know,” said Williams in a reference to the pulmonary embolism from which she suffered in 2011, threatening her life as well as her career.
“I have been through so much and I never thought I would be here,” she repeated. The victory, which was due to her very special competitive spirit and instinct for finding a way when no clear direction is evident, earns her the top spot at an age six months older than her fellow American Chris Evert did at 30 years and 11 months.
That was more than 27 years ago, underlining Williams’ status as one of the all-time greats, and possibly the finest woman player there has ever been.
Kvitova heaped praise on the American too. “I am glad she is No 1. She deserves it,” said Kvitova.
Williams now has a semi-final with Maria Sharapova, who has won the Qatar Open twice and remains unbeaten in Doha after winning 6-2, 6-4 win against Sam Stosur.
Victoria Azarenka moved closer to a second successful title defence in a row when she beat Sara Errani 6-2, 6-2 in the third quarter-final. She meets Agnieszka Radwanska in the other semi-final. Radwanska beat Caroline Wozniacki 6-2, 7-5 in the fourth quarter-final.