Trailing by a set and 4-0, Lapthorne turned the match around to earn a 2-6 7-5 6-2 victory in Nottingham.
Gordon Reid’s bid to reach the singles final ended with a 6-2 3-6 6-2 defeat by world number one Gustavo Fernandez.
Jordanne Whiley reached the women’s doubles final with Japan’s Yui Kamiji.
Whiley and Kamiji won 6-3 6-1 against another British and Japanese pairing in Lucy Shuker and Momoko Ohtani.
Whiley and Kamiji will meet Dutch top seeds Diede de Groot and Aniek van Koot – who they lost to in the Wimbledon semi-finals – for the trophy on Saturday.
World number three Lapthorne plays Niels Vink of the Netherlands in his final on Sunday after his extraordinary comeback in Friday’s semi-final.
“Heart and desire is what it’s all about this week,” Lapthorne said.
“As long as I compete well, that’s all I can do. I love winning, I love competing, I love playing those kind of matches.
“Playing matches like that, against a guy that pushes me to my limit and pushes me to dig deep within myself is the reason I play the game.”
Despite his singles disappointment, Reid has another chance to win silverware when he partners compatriot Alfie Hewett in the men’s doubles final on Saturday.