In the rain-hit Basque country Ander Herrera opened the scoring for Athletic only for Cesc Fabregas to pull Barca level on 24 minutes Sunday.
When Barca defender Gerard Pique put the ball into his own net with 10 minutes remaining, Athletic looked all set to hand the European champions their first league defeat of the season.
In a frantic finish to the match, however, Messi made the most of a defensive mistake to pounce on a loose ball and score his 14th in the league and first away from the Nou Camp this season.
Earlier, Real Madrid had a much easier task when Cristiano Ronaldo's hat-trick was the centre piece of a 7-1 hammering of helpless Osasuna, who have already been beaten 8-0 by Barcelona in mid-September.
It means Real, on 28, lead Barcelona (25) by three points with Valencia (24) in third a point further adrift.
The rain had fallen all day in Bilbao but the conditions did not affect the quality of football as both sides looked for the win from start to finish.
Despite dropping points, Barca coach Pep Guardiola was quick to hail Athletic's efforts.
"We've never played against such an energetic team, it was an extremely demanding and exciting match, a fantastic show put on by the 22 players," he said.
"When two teams want to win so much, football can be beautiful and the spectator is the winner."
Barcelona goalkeeper Victor Valdes had already been called into action by the time Herrera opened the scoring on 20 minutes with a curling left-footed shot after great work from Markel Susaeta down the left.
The advantage lasted only four minutes, Fabregas getting on the end of a left-wing cross from Eric Abidal to equalise with a looping header.
Barca looked the more likely winners with Messi and Andres Iniesta going close before Bilbao got their second.
It came with ten minutes remaining from an unfortunate mix-up in the Barcelona defence at a Bilbao corner. When Abidal tried to clear the ball it rebounded off Fernando Llorente before then hitting Pique and the back of the net.
A similar lack of clarity in the Bilbao defence gave Messi a chance to even the game and the Argentine was well placed to drill the ball low into the corner of the net to salvage a point for the Catalans.
Real's drubbing of Osasuna made it ten straight wins for Jose Mourinho's side, with Gonzalo Higuain, Pepe and two from substitute Karim Benzema adding to Ronaldo's hat-trick.
Osasuna had levelled through Ibrahima Balde early in the match but had Eneko Satrustegui sent off for a second yellow card in the 54th minute with the score already at 5-1.
For Mourinho, meanwhile, their winning run is fully deserved.
"We have deserved these 10 victories. We have not got them through luck, we have had to suffer for them like we did in Anoeta (Real Sociedad's stadium), but we've played either good or very good in all of them," he said.
Despite Ronaldo's hat-trick, his league tally of 13 means he trails top scorer Messi by a goal.
In the late game on Sunday Diego Castro converted an 82nd penalty to give Getafe a
3-2 victory over Atletico Madrid which lifted them from the bottom of the table to 14th.
An eventful derby match saw Getafe play with ten men for 73 minutes after an early red card for Alberto Lopo, but Atletico were always chasing a game they never looked capable of controlling.
Rayo Vallecano beat Real Sociedad 4-0, the losers playing for 79 minutes with 10 men after Inigo Martinez received a red card. Sociedad are now seven games without a win and replace Getafe at the bottom.
Zaragoza and Sporting Gijon also shared four goals in a bottom of the table clash, while two other games ended in scoreless draws on Sunday. Neither Racing Santander and Granada or Espanyol and Villarreal could find a goal between them.
On Saturday, Valencia beat city rivals Levante 2-0 away from home to leap-frog their neighbours into third place. A Javi Venta own-goal and a Tino Costa free-kick were enough for Valencia to edge a typically bruising derby match.
