The 20-year-old American, who took a set off Novak Djokovic and rattled the world number one at the US Open, is shaping up as potentially his country's biggest hope for the future in the men's game.
Californian Brooksby crushed Spain's world number 44 Alejandro Davidovich Fokina 7-5 6-0, and second seed Schwartzman awaits after the Argentinian took down another 20-year-old American, Brandon Nakashima, scoring a 6-4 6-2 quarter-final win.
Schwartzman is the man who ended Andy Murray's hopes at the tournament in Antwerp.
Brooksby, who had to win two qualifying matches to reach the main draw, said: "Tomorrow will be a great match. I'm really looking forward to it. I'll get some rest tonight and get ready to battle again tomorrow."
The other semi-final will see Italian top seed Jannik Sinner take on South African seventh seed Lloyd Harris following their respective straight-sets wins over Arthur Rinderknech and Marton Fucsovics.
There will be an all-Russian semi-final at the Kremlin Cup in Moscow, after Aslan Karatsev and Karen Khachanov won through to the final four.
Second seed Karatsev sank the hopes of Frenchman Gilles Simon, breaking serve five times on the way to a 6-4 6-3 win, while third-seeded Khachanov edged out Australian John Millman. A 7-5 7-6 (7-4) grind featured 19 aces by Khachanov, the 2018 Kremlin Cup champion, who saved four set points in the second set.
Marin Cilic thrashed Pedro Martinez 6-1 6-2 to set up a tussle with Ricardas Berankis, the Lithuanian ending Adrian Mannarino's run with a 6-2 7-6 (7-1) success. Frenchman Mannarino had stunned Russian top seed Andrey Rublev in the previous round.
A day after first and third seeds Fabio Fognini and Karen Khachanov were knocked out, second seed Shapovalov was beaten 7-5 6-4.
Humbert, who broke serve in the final game of each set, described the win as "one of the biggest of my career" as he set up a semi-final against John Isner.
The American was taken to two tie-breaks by Kyle Edmund but prevailed 7-6 (7-5) 7-6 (7-5), his 25 aces getting him over the line as he seeks a third title in Auckland.
Benoit Paire reached his first semi-final at the event, coming from a set down to beat John Millman 3-6 6-1 6-4.
The fifth seed will face Hubert Hurkacz, who spurned six second-set match points in a mammoth tie-break before eventually seeing off Feliciano Lopez 6-4 6-7 (11-13) 6-4.
At the Adelaide International, home favourite Alex Bolt was soundly beaten by Felix Auger-Aliassime, the Canadian needing just 55 minutes to win 6-3 6-0.
He will meet Andrey Rublev in the last four after the Russian battled past Dan Evans 6-4 3-6 6-3 in just over two hours.
Spanish hopes were extinguished in the quarter-finals, where Tommy Paul surprised Albert Ramos-Vinolas to win 6-3 6-4 and Lloyd Harris beat fourth seed Pablo Carreno Busta 6-3 6-3.
French Open quarter-finalist Sinner took just 61 minutes to seal a 6-1 6-2 victory against Duckworth, who replaced Hubert Hurkacz after the Pole withdrew due to food poisoning.
The 19-year-old Italian reached a career high 46 in the rankings following his exploits at Roland Garros and cruised past Duckworth after fending off two break points in a 12-minute game at 2-2 in the second set.
Sinner will go up against qualifier Pierre-Hugues Herbert in round two after the Frenchman defeated Tennys Sandgren 6-3 3-6 6-3 at the ATP event on Tuesday.
Zverev defeated Felix Auger-Aliassime in the final of last week's tournament in Cologne and will begin his bid for a second straight title against Millman after the Australian overcame Fernando Verdasco 6-4 6-2.
At the European Open in Antwerp, Alex De Minaur came from a set down to defeat veteran Richard Gasquet 3-6 7-6 (7-5) 6-3.
When serving for the set in the second set, Gasquet produced an unforced error following a reactive backhand from De Minaur. He then double faulted at 5-5 in the tie-break and the Australian eighth seed dished out the ultimate punishment.
Federico Coria stepped in after Kei Nishikori withdrew due to a right shoulder injury and the Argentinian was comfortably beaten 6-2 6-3 by Pablo Andujar.
The top seed won just two points across the first three games but took a first set that featured four breaks and lasted over an hour.
Duckworth broke in the first game of the second but eventually ceded that advantage and the frustrated Australian could not prevent Sinner from breaking again to claim a 7-6 (7-4) 6-4 win to set up a last-four clash with Filip Krajinovic, who prevailed in three sets against Kamil Majchrzak.
Gael Monfils will feature in the other semi-final after brushing aside Gianluca Mager 6-2 6-2, booking a meeting with American Marcos Giron.
It is a second semi-final in as many weeks for Monfils, who also reached the last four in Metz.
Giron will contest his first Tour-level semi-final following a 6-4 6-2 defeat of John Millman.
"Nothing is easy," Monfils said. "We have been putting in work for nearly four days here. Working on my serve and my forehand and it worked perfectly today. I will be ready for Marcos. He has been playing very well, so I think it will be a tough semi-final."
Verdasco, the ninth seed at the ATP 500 tournament, was beaten 6-3 6-3 by fellow Spaniard Pablo Andujar on the clay in Brazil.
The 36-year-old has fallen to three straight losses, the first of which was a third-round defeat at the Australian Open.
Verdasco – the 2018 runner-up – was one of three seeds tested, but Garin survived a test against Andrej Martin.
Garin, who won the title in Cordoba earlier this month, needed more than three hours to edge Martin 4-6 7-5 7-6 (7-5).
Albert Ramos-Vinolas departed, the seventh seed falling to 16-year-old Carlos Alcaraz 7-6 (7-2) 4-6 7-6 (7-2) after three hours, 36 minutes.
Federico Delbonis advanced, while wildcard Thiago Seyboth Wild needed almost four hours to get past Alejandro Davidovich Fokina 5-7 7-6 (7-3) 7-5.
There were also upsets at the Delray Beach Open, where John Millman and Adrian Mannarino were beaten in the first round.
Millman, the fifth seed, went down to Yoshihito Nishioka 3-6 6-4 6-2 and seventh seed Mannarino fell to Kwon Soon-woo 1-6 6-3 6-2.
Frances Tiafoe, the champion in 2018, came through his opener against qualifier Emilio Gomez.
