Lionel Messi came off the bench to set up a late equaliser for fellow substitute Munir El Haddadi, cancelling out Oscar De Marcos's first-half opener as Athletic were denied a first win at Barca's home ground since 2001.
However, Barcelona are now winless in their last three games - their longest run since December 2016 - and could surrender top spot later this evening when Real Madrid take on Atletico Madrid in the derby at the Bernabeu.
Despite Barca having dropped five points in their two games prior to this one, manager Ernesto Valverde opted to rest the likes of Messi and Sergio Busquets ahead of their Champions League trip to face Tottenham Hotspur next week.
The early signs suggested that the duo's absence would not be felt, with Philippe Coutinho being denied an early penalty shout after good link-up play with Luis Suarez and Nelson Semedo then dragging an effort wide moments later.
With Messi on the bench, Suarez got a rare opportunity to take on the free-kick duties and he flashed a fine 35-yard effort inches wide of the target as Barca looked to make a quick statement following their midweek defeat.
However, Athletic soon began to grow into the game and they almost broke the deadlock on 12 minutes when a simple pass forward sent Inaki Williams through on goal, only for the striker to see his lobbed effort drop narrowly wide with Marc-Andre ter Stegen beaten.
Ter Stegen then tipped a powerful Raul Garcia drive over the crossbar before the former Atletico Madrid midfielder turned provider, sliding a pass through for Williams, who took the ball around Ter Stegen but was denied a simple finish into the empty net by Arturo Vidal's tracking back.
The glaring miss almost proved costly when Barcelona created their standout chance of the half after 25 minutes, with Vidal slicing open the Athletic defence with an incisive through-ball for Suarez, who was denied on the turn by Unai Simon.
Ivan Rakitic then dragged an effort well wide before Suarez tested Simon again from a tight angle, and the resulting corner eventually fell to Coutinho, who worked space to shoot but then sliced his left-footed strike high and wide.
Athletic's opening goal did not come hugely against the run of play, though, as Barcelona were punished for more lacklustre defending when Markel Susaeta was given far too much time to lift a cross into the box for De Marcos, who ghosted in unmarked at the back post to cushion his volley past Ter Stegen.
It was a goal which threatened to plunge Barca closer to crisis, but they almost responded immediately when Suarez clipped a cross into the box for an unmarked Coutinho, who could not get enough on his header to beat Simon.
Coutinho had another sight of goal right on the stroke of half time when his scuffed volley bounced up and was tipped over the bar, but Barca left the field to jeers from the Camp Nou crowd at the interval.
There were no changes at the break, but Busquets and Messi were both introduced within 10 minutes of the restart and the latter's appearance sparked the beginning of a growing period of pressure from the home side.
Athletic needed the woodwork to rescue them twice in the final half-hour, with the first of those instances seeing Coutinho's volley bounce up against the crossbar and back out towards Suarez, who could not contort his body enough to convert the rebound.
Messi saw a free kick cleared off the line 10 minutes later, but Athletic offered a warning shot that they could still pose a threat on the break moments later when it took a fine last-ditch challenge from Semedo to deny Williams.
Such opportunities were few and far between for the visitors in the second half, though, and they were thankful to the frame of the goal again with 13 minutes remaining when Messi struck the post after a one-two with Jordi Alba, before curling another effort wide moments later.
The equaliser finally arrived with six minutes remaining of normal time, and it was no surprise to see Messi heavily involved as he saw one shot saved before latching on to the rebound and firing it across goal for Munir to steer home from close range.
Barca almost won it late on too when Messi clipped a pass into the path of Rakitic, but the Croatia international lifted his finish over the crossbar as Valverde's side dropped yet more points in a miserable week for the club.
The champions now sit one point above Real Madrid and three clear of Atletico Madrid ahead of this evening's derby between the city rivals, while Athletic are still winless since the opening day but have now held Barca and Real to draws in that time.
BARCELONA (4-3-3): Ter Stegen; Semedo, Pique, Lenglet, Alba; Roberto (Busquets 51'), Vidal (Messi 55'), Rakitic; Coutinho, Suarez, Dembele (Munir 80')
ATHLETIC BILBAO (4-2-3-1): Simon; De Marcos, Alvarez, Martinez (Nolaskoain 23'), Balenziaga; Benat (San Jose 67'), D Garcia; Susaeta, R Garcia, Yuri; Williams (Aduriz 79')