Midfielder Mikel Rico opened the scoring with just six minutes played and Ander Herrera doubled the lead just four minutes later.
Aymeric Laporte made it 3-0 just after the half-hour mark, but Helder Barbosa gave Francisco Javier Rodriguez's side faint hope by scoring before half-time.
However, further goals from Aritz Aduriz put the hosts further ahead and a double from substitute Ibai Gomez completed the rout.
Athletic had failed to score in either of their opening two games of 2014, 1-0 defeats at Real Sociedad and Real Betis, but needed just six minutes to find the net against Almeria.
Rico released Aduriz down the left hand side of the area and the striker pulled a pass back across the area for the midfielder to meet and drill a left-footed shot into the top corner.
Athletic scored again just three minutes later.
Markel Susaeta dispossessed left-back Sebastian Dubarbier in his own area and found Herrera at the far post. The midfielder controlled the ball before shooting past Esteban, the ball bouncing off the goalkeeper on its way in at the near post.
Athletic continued to dominate and the third goal arrived when Laporte headed a corner from Susaeta into the bottom corner of the goal, just past an outstretched Esteban.
Almeria, normally an adventurous, attacking side, struggled to create chances, but pulled a goal back from a set-piece.
Verza whipped in a free-kick from the left-hand side and Barbosa rose above the defence to flick a header into the far corner.
Athletic had three occasions to restore their three goal advantage before the break, Aduriz, Herrera and Ander Iturraspe all coming close but failing to find the net.
The half-time break provided little respite for the visitors, as Aduriz got off the mark seven minutes into the second half.
Herrera released Ibai down by the byline and the winger put the ball into the box towards Muniain, who was unable to reach it, so Aduriz fired home instead.
Ibai, who had only entered the game on 50 minutes to replace Susaeta, soon added goals to his assist.
The first came with 22 minutes remaining. The winger received the ball on the edge of the area and, faced with little space, switched to his right foot and smashed an unstoppable effort past Esteban.
With five minutes remaining substitute Kike Sola was pushed over in the box by Angel Trujillo and Ibai stepped up to shoot beyond Esteban, the goalkeeper guessing the right way.