An entertaining game played at frenetic pace was decided in the eighth minute when Laporte met Markel Susaeta's corner and powered the ball past Getafe goalkeeper Miguel Angel Moya.
The home side should have levelled from the penalty spot shortly before halftime after Adrian Colunga was barged to the ground by Carlos Gurpegi but Getafe midfielder Diego Castro's spot kick was brilliantly saved by Gorka Iraizoz.
It was Athletic's first away win in four attempts since an opening day success at Real Valladolid with midfielder Ander Herrera saying the Basque club had banished the fragility that led to three straight defeats on the road.
"Today it wasn't like that and we didn't allow them many chances while not creating all that many for ourselves," Herrera told Spanish television broadcaster Cuatro.
"We have to challenge ourselves and I think we have the squad for that but we also have to be careful as the season is very long and any team can surprise you," he added.
Castro, who had scored all 15 of his previous penalties in his La Liga career, said a draw would have been a fairer result.
"It was a combination of a lack of precision and a bit of bad luck in front of goal," he said.
Bilbao have 19 points from 10 matches, one fewer than fourth-placed Villarreal, with Getafe on 16 points in sixth.
Unbeaten leaders Barcelona with 28 points pulled six clear of Real Madrid when the champions beat their arch rivals 2-1 in Saturday's 'Clasico' at the Nou Camp.
Atletico Madrid are a point behind Barca in second after their 5-0 drubbing of Real Betis at the Calderon on Sunday.
There is a midweek round of La Liga games this week including Barcelona at Celta Vigo on Tuesday. Real host Sevilla on Wednesday and Atletico visit Granada on Thursday.
- Sports & Recreation
- Markel Susaeta
- Real Valladolid
- La Liga
- Carlos Gurpegi
- Ander Herrera