Aritz Aduriz saw his 96th-minute penalty saved by Manolo Reina, after Abdon Prats had dragged a spot-kick wide for the hosts.
The point keeps Bilbao unbeaten after four league games this season, but victory would have seen them move two points clear of Atletico Madrid before the early leaders' game at Real Sociedad on Saturday.
The 38-year-old Aduriz had scored a stunning winner as a substitute against reigning champions Barcelona in the opening match of the season, but he turned villain in Mallorca as Reina dived to his left to palm away the striker's penalty.
In the 82nd minute, Prats, who also came off the bench, had missed Mallorca's effort from 12 yards.
The drama was not over there, though, as Mallorca were not given another spot-kick following a VAR review, before they had a goal ruled out for offside in the ninth minute of added time.
On Saturday, Barcelona host Valencia, who surprisingly sacked coach Marcelino earlier this week, after Real Madrid take on Levante.