\Teenager Muniain netted what would prove to be the deciding goal in the 40th minute, not long after Slovan defender Karim Guede had cancelled out Markel Susaeta's opener in the Group F clash at the Pasienky stadium.
The victory is the first Athletic have earned in competition since former Argentina coach Marcelo Bielsa took over this summer, with the Basque side having taken just just one point from their first two Primera Division matches.
Athletic were also held 0-0 at home by Trabzonspor in their Europa League play-off despite the Turkish side playing with a man less from the sixth minute.
Athletic did not play what could have been a tough return match as Trabzonspor were promoted into the Champions League following the withdrawal of Fenerbahce, leaving the Spanish side to automatically progress through to the Europa League group stages.
Bratislava, who took the prized scalp of Roma in the play-offs, started well and almost took a second-minute lead through Igor Zofcak, whose 25-yard free-kick was parried away by Gorka Iraizoz.
However, 10 minutes later Athletic were in front when Susaeta slotted home from Carlos Gurpegi's pass.
Igor Gabilondo then sent a volley wide for the visitors before Slovan drew level in the 34th minute, Guede the goalscorer.
Parity lasted only six minutes, though, before Athletic regained their lead when 18-year-old Muniain netted a fine goal, the Spain Under-21 international volleying into the roof of the net after collecting Gabilondo's left-wing cross.
That was the score as the teams went into half-time, but the home side came close to equalising immediately after the break when Marko Milinkovic saw a header from a corner blocked by Fernando Amorebieta.
Slovan would not come any closer to drawing level after that, though, as the Athletic defence held firm to help their side claim the points.