Tirante, ranked 83rd in the world, notched his second career win over a top-10 player as he sent the ninth-ranked Shelton packing to reach the second ATP semi-final of his career.
"I knew that Ben was a very difficult player, a great player, so I had to take more risks at some times of the match," said Tirante, who fended off a break point early in the third set and broke Shelton for a 5-4 lead before serving it out with a comfortable hold.
"I did sometimes good, I did sometimes bad, but that's the key. (I had to stay) mentally strong all the time and try to break the serve - he serves amazing."
Burruchaga, ranked 77th, upset third-seeded American Learner Tien, ranked 22nd in the world, 7-5, 6-4 to reach his first career semi-final.
The son of former soccer player Jorge Burruchaga, who won the World Cup with Argentina in 1986, the 24-year-old had already knocked out another member of the world top 40 on Thursday, 33rd-ranked local favourite Brandon Nakashima.
Second-seeded American Frances Tiafoe saved a match point in the third set tiebreaker to reach the semi-finals with a 3-6, 6-4, 7-6(6) victory over Australian Alexei Popyrin.
Tiafoe will face fourth-seeded Tommy Paul in an All-American semi after Paul beat Argentina's sixth-seeded Tomas Etcheverry 6-4, 6-2.
