Beaumont secured 12 of the 26 votes needed to prevent Lapasset remain in position, with the Frenchman picking up 14 votes for a tiny majority.
The ex-England lock, who captained the Lions in South Africa back in 1980 and was team manager in 2005, also lost his role as vice chairman in the process.
Beaumont had the support of all four Home Unions but Lapasset's success in getting Rugby Sevens into the 2016 Olympic Games proved to be decisive as it secured him the backing of the majority of the developing nations.
Lapasset's new four-year term will begin on January 1, as will that of the new vice chairman Oregan Hoskins of South Africa.
Hoskins narrowly prevented Beaumont from retaining his previous role as Lapasset cast the all-important vote after the two men had been tied at 13 votes apiece.