Alcott, who beat the Briton to win the Wimbledon title in July, prevailed 0-6 7-6 (7-3) 6-3 in New York.`
Lapthorne defeated David Wagner on Thursday and will play American Bryan Barten in his final round-robin match.
The top two after the completion of the round-robin tournament will advance to Sunday’s final.
Lapthorne will also team up with Alcott in Saturday’s quad doubles final, having won together at Wimbledon.
Elsewhere at Flushing Meadows, defending champion Alfie Hewett’s wheelchair singles quarter-final against Japanese second seed Shingo Kunieda was suspended because of the poor weather in New York on Friday, with the Briton leading 4-1 in the first set.