World number five Whiley, from Ickenham, came through her quarter-final against Kaitlyn Verfuerth in straight sets.
Whiley, who will also be looking to defend her doubles title at Flushing Meadows, traded breaks early on against the 30-year-old American before winning the first set 6-3.
She then raced through the second set 6-1 to set up a semi-final against Dutch second seed Aniek van Koot.
Eastcote’s Lapthorne had lost to world number one Dylan Alcott on Thursday in the first match of the four-player round robin in the quad singles event.
However, he recovered to beat the USA’s Nick Taylor, the world number 10, in three sets.
Taylor, who at 35 is 11 years older than Lapthorne, took the first set 6-3, but the west Londoner responded to pinch the next two sets 7-5.
World number four Lapthorne will now need to beat David Wagner, ranked number two, and hope Alcott loses to Taylor.
After his match, Lapthorne tweeted: “Won today 3-6 7-5 7-5 need to win tomorrow and pray for a miracle, gutted about how I’ve been mentally.”
Won today 3-6 7-5 7-5 need to win tomorrow and pray for a miracle, gutted about how I’ve been mentally. Doubles is very important now — andy lapthorne (@lapstar11) September 11, 2015