Bilbao opened the scoring in the ninth minute when Etxeita headed into the net from a corner.
Rico did double their lead in the 26th minute, chipping the ball over the goalkeeper from outside the penalty area.
Athletic continued to dominate with Rico, Iker Muniain and Ibai Gomez all coming close before the break.
Almeria came out for the second half strongly and pulled a goal back in the 47th minute through Wellington Silva, whose shot from outside the penalty area found its way through the Bilbao defence and into the net.
Inaki Williams could have extended the lead again for Bilbao, but he headed over the bar on the hour mark before a quiet finish ensured that Athletic took the points.
The result means that Almeria remain level on points with the relegation zone, while Bilbao stay in eighth.