Los Leones had not beaten their Madrid-based opponents in five matches and had never won in the capital in nine meetings between the teams before Monday night's encounter, but Laporte's ninth-minute header brought that run to an end.
Getafe midfielder Diego Castro wasted an excellent opportunity as his spot-kick on the stroke of half-time was expertly saved by Bilbao goalkeeper Gorka Iraizoz, whose side moved up to fifth in the standings following the 1-0 victory.
Iraizoz was called into action in the first few minutes, expertly parrying away Pablo Sarabia's shot from inside the area.
It was to prove a pivotal save as Laporte opened the scoring for Bilbao in the ninth minute.
The 19-year-old French defender, who has been capped at under-21 level, rose highest to head into the top corner from Markel Susaeta's corner from the right.
Roberto Lago wasted a chance to draw the hosts level when he missed the target with his strike from a decent position, while at the other end Ander Herrera's long-range shot kept Miguel Moya on his guard.
Much-travelled striker Adrian Colunga headed over the bar for Getafe, who were finding it increasingly difficult to impose themselves in the final third.
Yet just before half-time, it appeared they were given a lifeline when Bilbao midfielder Carlos Gurpegi fouled Colunga in the area, with referee Jesus Gil wasting no time in pointing to the spot.
However, Diego Castro's placed penalty to the bottom right was magnificently tipped away by Iraizoz to keep the Basque club's lead intact.
Getafe were undeterred, though, and Diego Castro shot wide from a good position, while Pedro Leon's shot from just outside the area was well saved by substitute Bilbao goalkeeper Iago Herrerin, who had replaced Iraizoz at the break.
Colunga also spurned a decent opportunity by missing the target although only a fantastic save from Moya prevented the visitors from compounding their woes as the Getafe keeper parried away Ander Iturraspe's 20-yard drive.