Goerges also battled back from a set down to beat Dutchwoman Kiki Bertens 3-6 7-5 6-1 in the last of the women's quarterfinals.
Bertens, who was contemplating retirement previous year after losing her enthusiasm for the sport, is the first Dutch woman to reach the Wimbledon quarter-finals since Michaella Krajicek in 2007.
Williams is into the 35th Grand Slam semifinal of her career and her first since winning last year's Australian Open. From then on, the result felt inevitable as she served beautifully, levelling the match and then, after a stunning backhand to break for 2-1 in the decider, she eased through the set, serving out to love to take her place in the last four.
Williams hit six of her seven aces in the final set, including one clocked at 122 miles per hour, and won 44 of the last 54 points she served. Be a good friend and let her win'.
Through her four Wimbledon matches last week, Williams carefully had calibrated the speed of her serves, taking care not to aggravate the injured pectoral muscle that had forced her withdrawal midway through the French Open last month. Plus, at Wimbledon in particular, she lost in the first round each of the past five years.
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Giorgi, playing in her maiden major quarterfinal, threatened an upset by winning the first set, and led 30-0 on Williams' serve at 1-1 in the second.
Serena is the first woman to reach a Grand Slam semifinal without facing a top 40 ranked opponent since 2013. It was a fearless performance that offered Williams one of her toughest tests since returning.
"They know they have to play their absolute best if they want to stand a chance against her - and hope that she has a really bad day", Keothavong, 34, said with a smile.
She went a couple of weeks without practicing serves, and the rest did wonders.
The ICC shared an image of the Test rankings for batsmen which showed the 20-time Grand Slam victor as the number one batsman in the world. "I'm just here just to be here and to prove that I'm back".
However, Williams lifted her intensity to level, before closing out the match to move a step closer to reclaiming the title she won in 2016.
Hsieh protested, leading the match referee to come on to discuss the decision with the umpire as calls of "replay the point" came from a bemused crowd on Court 18.