Athletic's new Argentine coach Marcelo Bielsa had been under pressure after the Basque club took only two points from their opening five games but a second consecutive win lifts them to 11th on eight points from seven matches.
Spain Under-21 international Muniain (pictured) put the home side ahead on the half hour at their imposing San Mames stadium, before Gabilondo and Martinez made it 3-0 before the break.
Osasuna's Iran midfielder Javad Nekounam pulled a goal back from the penalty spot in the first minute of second-half stoppage-time after Martinez handled the ball in the area.
The Pamplona-based club finished the match a man down after their Ivory Coast-born midfielder Roland Lamah earned two yellow cards in the space of seven second-half minutes.
The second came with around 20 minutes left after he appeared to catch Muniain on the side of the head with a flailing arm.
In weekend action, La Liga top scorer Lionel Messi netted his ninth and 10th goals of the campaign in a 3-0 win for Barcelona at home to Racing Santander on Saturday.
The champions were joined at the top on 17 points by surprise package Levante. The tiny Valencia-based club extended their remarkable early-season run with a 3-0 win at home to 10-man Malaga on Sunday.
The pair have a one-point advantage over third-placed Real Madrid, who thumped visiting Real Betis 4-1 on Saturday thanks to a third hat-trick in two weeks from Messi's Argentina team-mate Gonzalo Higuain.
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