'I struggled to win my first WTA title. It took me time. So I believe this will take me time,' said the Tunisian.
It was twice French Open champion Swiatek's first major title on a hard court and the 21-year-old is the first Polish woman to win the US Open.
After a sluggish start, the Pole came alive in the second set and overcame an early break in the decider by winning the final four games to see off the powerful Belarusian.
It will be her second straight US Open semifinal after a convincing 6-1 7-6(4) win over the Czech star.
Alcaraz, last year's quarterfinalist, remained on course for another memorable run at Flushing Meadows after squeezing past 2014 champion Cilic to set up a showdown against Jannik Sinner.
Second seed Anett Kontaveit also had her preparations for Flushing Meadows cut short in the round of 16.
The favourtie, on a 37-match winning streak, was stunned by France's Alize Cornet 6-4 6-2.
The World No 1 defeated Dutch player Lesley Pattinama Kerkhove 6-4, 4-6, 6-3.
The Spaniard beat Francisco Cerundolo 6-4 6-3 3-6 6-4 in the first round.
Djokovic and Nadal find themselves in different halves of the draw and they cannot face each other until the men's final on July 10.