Mikel Rico, Ander Herrera, and Aymeric Laporte all scored first half-goals before Helder Barbosa pulled one back for the visitors in the 34th minute.
But Bilbao continued to dominate in the second half at its new San Mames Stadium as Aritz Aduritz restored the three-goal cushion before Ibai Gomez scored a brace, including a late penalty.
Athletic enjoys a four-point cushion in fourth at the halfway point of the season, while Almeria sits three points above the drop zone.
Atletico Madrid's title credentials get a big test later Saturday when it welcomes Barcelona to the Vicente Calderon Stadium in a matchup between the co-leaders.
Barcelona and Athletico Madrid were satisfied with a share of the spoils after they cancelled each other out in an intense 0-0 draw.
The stalemate at the Calderon moved the joint La Liga leaders on to 50 points at the midpoint of the season.
Third-placed Real Madrid will close to within three points if they can beat mid-table Espanyol on Sunday (1800 GMT).