The top seeds defeated Australia’s Dylan Alcott and South Africa’s Lucas Sithole 6-0 3-6 6-2.
“It’s an amazing feeling and a lovely end to a tough couple of days,” said 24-year-old Lapthorne.
Meanwhile, fellow Britons Jordanne Whiley and Gordon Reid both reached their respective doubles finals.
Whiley and Japanese partner Yui Kamiji stayed on track to retain their women’s doubles title after the Grand Slam-winning pair defeated Germany’s Katharina Kruger and Dutchwoman Sharon Walraven 6-4 6-3.
In Friday’s final they will face Dutch second seeds Jiske Griffioen and Aniek van Koot who they beat in the Roland Garros, Wimbledon and US Open finals last season, although the Dutch pair won their most recent encounter in the Sydney International Open.
“I’m really happy to make the final, but it wasn’t my best day,” said Whiley. “I certainly need to play better in the final.”
Reid will contest his second Australian Open men’s doubles final after partnering Argentina’s Gustavo Fernandez to win their semi-final against second seeds Joachim Gerard of Belgium and Maikel Scheffers of the Netherlands 6-1 7-5, having come from 2-0 down and 5-3 down in the second set.
“I’m delighted to be back in the final after a good solid performance from us – we’re pleased how we battled through the second set,” said Reid, who lost in last year’s final.
Reid and Fernandez will play defending champions and top seeds Stephane Houdet of France and Shingo Kunieda of Japan in Friday’s final.
Earlier on Thursday, Lapthorne saw his hopes of a place in the quad singles final fade when he was edged out 6-3, 7-6 (7-4) by Sithole in his second of three round-robin singles matches.