Ter Stegen turned away Markel Susaeta's arcing shot before the break and then produced a superb reflex stop to repel Inaki Williams' effort from point-blank range as time ran down on a night when the La Liga leaders were unable to restore their eight-point advantage over second-placed Real Madrid.
Barcelona have dropped four of the last six points available, failing to capitalise on Atletico Madrid's two defeats and have therefore given Real Madrid reason to believe they can pull off a miraculous comeback in the title race.
"We were first last week and we are still first, and if my maths is right, we will be again next week", Valverde said. Messi was picked but not fully recovered while Ousmane Dembele was fit enough only for the bench, having shaken off an ankle injury.
The visitors eased themselves into the game as Messi and Suarez, prompted by Philippe Coutinho, started to make their presence felt, although Athletic were handed the 11th-minute opportunity when Yuri Berchiche intercepted Arturo Vidal's loose pass and raced forward before firing just wide.
Williams had a wonderful chance to send Bilbao ahead in the 80th minute when the forward met a free kick from Muniain, but the ball flew just wide of the post, before the in-form Spain global fired straight at Ter Stegen from another Muniain pass moments later.
Lionel Messi had Barca's best chance as his lob from outside the box hit the top of the crossbar. Barca enters the game in first place with 50 points and a record of 15-5-2, while Bilbao is in 13th place with a record of 5-11-6 and 26 points.
Substitute Mikel San Jose blasted just wide as the home side finally managed to put together a second-half attack with 65 minutes gone and Williams saw an audacious flick from De Marcos' cross blocked six minutes later.
Bilbao had De Marcos sent off for a second bookable offence - deliberate handball - in the final moments, but Barcelona, who were denied a late penalty by VAR, could not find a victor as they failed to triumph for the third match in succession.