The game was suspended on Sunday after 17 minutes when Athletic goalkeeper Unai Simon alerted the referee to a problem with a supporter at Estadio Los Carmenes.
Doctors from both teams rushed to help, but the fan - who had suffered a cardiac arrest - could not be resuscitated.
The teams left the field and agreed to continue the match on Monday.
Both clubs laid flowers on the fan's seat and held a minute's silence before restarting the match in the 17th minute with Athletic leading 1-0 thanks to a goal from Inaki Williams.
He almost got a second before halftime only to hit the bar, as the visitors controlled the game but were unable to take their chances.
They paid for their profligacy 10 minutes into the second period when Granada drew level when Bilbao defender Inigo Ruiz de Galarreta diverted an intended clearance into his own goal.
"We feel like we've lost two points," Williams told broadcaster Movistar.
"We were superior but we were not clinical. We go away frustrated but there's no time for lamenting."
Rayo Vallecano drew 0-0 at home to struggling Celta Vigo, who remain in the bottom three.
The hosts had all the chances and had a Florian Lejeune goal chalked off after 11 minutes as goalkeeper Vicente Guaita excelled.
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