Muniain's 32nd-minute strike saw Bilbao lead at the interval, but Isco levelled the scores in the 64th minute as both teams continued their unbeaten start to the 2018-19 campaign.
The result has left Los Blancos in second position in the table, two points behind leaders Barcelona. Bilbao, meanwhile, occupy fourth, and have a game in hand after their clash with Rayo Vallecano was postponed earlier this month.
A bright start from Madrid saw Luka Modric and Marcelo both have half-chances in the first two minutes, but the former saw his effort deflected wide before the latter missed the Bilbao post from distance.
The home side's first shot of the match arrived in the eighth minute, although the effort from Raul Garcia was always missing the target.
Raul Garcia then set up Inaki Williams for a brilliant chance in the 14th minute, but the striker's low effort drifted wide of the far post, before Markel Susaeta's powerful drive missed the Madrid goal moments later.
The visitors were also having chances though, with Karim Benzema's strike missing the crossbar in the 18th minute.
It was Bilbao that made the breakthrough in the 32nd minute, however, with Muniain bundling a low Oscar de Marcos cross into the back of the net. Williams actually looked to make first contact, but the final touch came from Muniain as Bilbao moved one goal ahead on the night.
Raul Garcia looked to have sent Bilbao 2-0 ahead in the spell of pressure that followed, but the midfielder's effort from a deep free kick was ruled offside.
Modric's powerful strike looked to be sending Real Madrid level in the 37th minute, but Bilbao goalkeeper Simon made an excellent low save to keep his team ahead, and 1-0 to the hosts was the score at the half-time break.
Real Madrid's first strike of the second half came from Bale in the 54th minute, but the Welshman's effort was always missing the goal.
Bale tested Simon from a long-range free kick just before the hour, but the goalkeeper was equal to it, and then recovered to prevent Sergio Ramos from converting the rebound.
Real Madrid were level in the 64th minute, however, when Isco - who had not long on the field - headed a Bale cross into the bottom corner of the net to silence San Mames.
Simon kept out a strike from Asensio in the period that followed, before Ramos curled a free kick over the crossbar 17 minutes from time as the away side continued to look for a second.
Marcelo was next to try his luck for Real Madrid in the 82nd minute, with the left-back missing the crossbar from inside the Bilbao box as the scores stayed level.
The final shot of the match came from Mikel Rico in the third minute of added time at the end of the 90, but Courtois was on hand to make the save as the points were shared at San Mames.
ATHLETIC BILBAO (4-3-3): Simon; De Marcos, Yeray, Inigo, Berchiche; D Garcia, R Garcia, Benat (Rico 81'); Susaeta, Williams (San Jose 76'), Muniain (Capa 54')
REAL MADRID (4-3-3): Courtois; Carvajal, Ramos, Varane, Marcelo; Modric (Isco 61'), Kroos, Ceballos (Casemiro 46'); Asensio, Benzema, Bale (Lucas 75')