Iker Muniain bundled home from close range to give Athletic a first-half lead in a bad-tempered clash at San Mames, while Unai Simon impressed in goal for the host.
But Toni Kroos sent Gareth Bale clear with an excellent chipped pass and substitute Isco flicked in the Wales winger's cross to level shortly after his second-half introduction.
Madrid pressed for a winner but could not beat Simon a second time, leaving Lopetegui's men two points behind league leader Barcelona, which maintained its perfect record.
Inaki Williams dragged a 14th-minute shot wide and Markel Susaeta came close with a 20-yard drive before twice being denied by Thibaut Courtois as Athletic made a high-octane start.
Simon saved from Marco Asensio, but Athletic took a merited lead in the 32nd minute.
Susaeta got into the Madrid box and the captain's low cross was turned in from close range by a combination of Williams and Muniain, the latter credited with the goal.
Athletic had a second ruled out for offside when Raul Garcia tucked home a Benat free-kick, with Simon again in action as he dived low to his right to keep out Luka Modric's deflected drive.
Simon was fortunate to get away with a poor parry from Bale's free-kick, but the 21-year-old brilliantly made amends to keep out Sergio Ramos on the rebound.
Lopetegui brought on Isco and the Spain star struck the equaliser within three minutes, nodding in unmarked to give Bale's excellent right-footed cross on the run the finish it deserved.
But Simon superbly blocked an Asensio shot in the 68th minute to preserve a point for Athletic, neither side able to find a winner in a frantic finale.
It is UEFA Champions League action next for Lopetegui's men, who host Roma before welcoming Espanyol to the capital in LaLiga three days later.
Athletic heads to Real Betis for its next league clash.