Nico, who impressed with his international debut for Spain last week, crossed the ball for older brother Iñaki to put Bilbao ahead in the 11th minute with a header.
After Oihan Sancet made it 2-0, it was Iñaki's turn to set up his younger brother in the 62nd.
Nico exchanged two quick passes with Iñaki inside Almería's area before he fended off defender Rodrigo Ely and poked the ball between the legs of goalkeeper Fernando Martínez.
It was a third goal in as many rounds for the 20-year-old winger, who may be heading to the World Cup after he set up the winner for Spain to edge Portugal 1-0 on Tuesday and clinch a spot in the Nations League final four.
Iñaki also debuted for Ghana, another World Cup team, last week after he changed allegiance from Spain.
"We are enjoying some incredible weeks in our family, we are very happy," Iñaki said. "The team has taken a step forward this season."
Mikel Vesga added a fourth goal from the penalty spot with six to play after he was fouled by Arnau Puigmal.
Bilbao has excelled since Ernesto Valverde returned to start his third stint as coach at the club.
A third straight win left Bilbao and Barcelona two points behind league leaders Real Madrid before both powerhouses play this weekend.
Barcelona visit Mallorca on Saturday, while Madrid host Osasuna on Sunday.
