The 24-year-old staved off match point as he produced a 7-6(5), 3-6, 7-6(6) win over fellow American Paul, ranked 13th in the world.
Brooksby has made a habit of saving match points in this tournament; he did so in the first round of qualifying and then fought off three match points in the second round of the main draw.
The match was blighted by several lengthy rain delays and concluded seven and a half hours after it began when Brooksby's backhand return winner secured the tie-break win in the third.
"I’ve been able to recall a lot of those (match point) moments this tournament and I think I've gotten stronger as each of those have gone on," said Brooksby who is ranked 50th in the world.
"I'm just really pleased with how I've kept my mental composure here the whole week so far," he said.
Brooksby will face second-seeded fellow American Francis Tiafoe in Sunday's final.
Tiafoe took down fourth seed Brandon Nakashima 6-4, 7-6(3) to reach the final.