Thirteenth seed Broady, from Stockport, beat Wayne Montgomery of South Africa 7-6(6) 7-6(4) in the quarter-finals - his third match in two days at Flushing Meadows due to the adverse weather this week in New York.
The 18-year-old will now play eighth seed Kaichi Uchida for a place in the final. Uchida of Japan put out top seed Kimmer Coppejans in the quarters.
Broady is hoping to emulate fellow Brit Oli Golding, who won the title last year.
In the boys' doubles, Britain's Kyle Edmund and partner Frederico Ferreira Silva of Portugal beat Yoshihito Niskioka of Japan and Jorge Brian Panta Herreros of Peru 6-2 6-4 to make the final.
They will now meet Nick Kyrgios and Jordan Thompson of Australia in Saturday's title match.