Marcelo Bielsa's men are now level on points with Espanyol in fourth spot after a storming display after the half-time interval
In a match with numerous chances at either end, Marcelo Bielsa's side proved the more clinical after the break. Fernando Amorebieta, Mikel San Jose, and Toquero struck within a five-minute period to secure the points.
Seven sides in La Liga can lay claim to a challenge for that elusive fourth spot and entry into the Champions League qualifiers, and Bilbao and Malaga both knew that a win would go some way to legitimising their own chances.
The open nature of the first 45 minutes highlighted the desire of both teams to secure the points, but a mixture of solid goalkeeping and poor finishing ensured that the match entered the interval level.
However, Bilbao emerged as the more urgent of the two sides in the second period, and a three-goal salvo around the hour mark proved the beating of Manuel Pellegrini's visitors.
On 57 minutes following an inswinging corner, San Jose's shot was blocked, only for Amorebieta to lash home the rebound with his left foot and Bibao had the lead.
The hosts were in complete control. Iker Muniain drew a solid save from Willy Caballero, before the second goal arrived to kill off the contest.
Toquero's delivery into the area found San Jose unmarked, and the former Liverpool defender powered home his header, as Malaga's defenders stood and watched, unable to keep him at bay.
Malaga were shellshocked, and a minute later, Bilbao were out of sight. Toquero, who enjoyed an excellent all-round afternoon, this time finished off a move himself, converting Ander's cross into the area from an acute angle past Caballero.
Pellegrini's side must have wondered how they found themselves in this situation, having been very much in the match up until half-time, and they attempted to fight their way back into the contest.
However, Joaquin and Salomon Rondon both wasted decent openings, and the Basque hosts were able to see out what could prove to be an invaluable three points come the end of the season.
The win sees Bilbao move level with Espanyol on 33 points, while Malaga, despite being in eighth position, are just two points further back.