Second seed Rafael Nadal – the winner at Flushing Meadows in 2010, 2013 and 2017 – produced some electrifying moments of brilliance to end the challenge of Croatia's Marin Cilic 6-3, 3-6, 6-1, 6-2.
Tsitsipas had never won a set in his previous three meetings against claycourt specialist and 11-times French Open champion Nadal, but the Greek drew first blood by saving a break point to hold in the 10th game and take the first set.
Marin Cilic beat Lucas Pouille in straight sets to claim Davis Cup for Croatia.
As the Spaniard overcame Marin Cilic it was the Greek teen Stefanos Tsitsipas who was the talking point after he defeated defending champion Alexander Zverev in three sets.
Romanian and world number one Simona Halep broke Spaniard Garbine Muguruza's serve six times in a ruthless display to set up Saturday's final against US Open champion Sloane Stephens.
Top seed Simona Halep faces Spain's Garbine Muguruza in the semifinals as the Romanian bids to claim her maiden Grand Slam title.
Nadal, still unbeaten in semi-finals at the Foro Italico, unleashed a trademark clay-court attack after winning a tight first set lasting well over an hour in the 51st clash between the two players.
Nadal and Djokovic have played each other 50 times, but their last meeting came more than a year ago, when Nadal won in the Madrid Open semifinals. Djokovic leads Nadal 26-24 overall, but Nadal holds a 13-7 advantage on clay.
After going four years without winning a major, Switzerland's Roger Federer has now won three of the last five.
Sixth-seeded Marin Cilic of Croatia meets Roger Federer in Sunday's final of the 2018 Australian Open, with the Swiss aiming for his 20th grand slam title.
Hyeon Chung is the first Korean to reach the last four of a Grand Slam. On the women's side, Germany's Angelique Kerber Kerber is the only Grand Slam singles champion still in the draw.
Six-time champion Novak Djokovic and top seed Rafael Nadal are both are out of the Australian Open tennis tournament after each losing their quarterfinal.
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