The decisive moment came in the seventh minute when the midfielder lashed the ball home following Pedro Leon's corner.
There were chances at either end to add to the scoring but nobody was able to match Borja's precision as Getafe held out for three points.
A corner after only two minutes almost yielded the breakthrough for the home side, Sergio Escudero denied when his long-range shot rebounded off the crossbar.
They did not have to wait long for their goal though, Leon curling one to the far post and Borja showing no doubt when the chance fell to him 12 yards out.
Athletic could not handle their opponents' set-piece delivery and yet another Leon corner two minutes later saw Rafa Lopez's header saved by Gorka Iraizoz.
Aritz Aduriz finally registered an attempt for the visitors when his low shot was blocked by the feet of goalkeeper Miguel Moya.
Indeed Aduriz had his side's first three chances, none successful.
The second period started less frenetically than the first, though Escudero did chance his arm from 25 yards in an attempt to push Getafe further ahead.
Speculative efforts seemed to be the order of the day now, Ander Herrera and Abdelaziz Barrada trading wayward shots around the hour mark.
Leon seemed to have more space than most and reeled off two shots on the Athletic goal as the time petered out but in truth a second goal never seemed overly likely.