Cristiano Ronaldo struck a hat-trick as Real Madrid crushed Osasuna 7-1 and Jose Mourinho's team lead the standings on 28 points, three ahead of the European and Spanish champions.
The clash between Barcelona coach Pep Guardiola and Argentine Marcelo Bielsa had been eagerly anticipated and produced a breathless encounter in Bilbao.
Barca goalkeeper Victor Valdes was beaten for the first time in 10 matches when Ander Herrera fired Bilbao in front in the 20th minute, but Cesc Fabregas headed the visitors level just four minutes later.
Standing water on the pitch added to the tension and the conditions contributed to both late goals.
A Bilbao corner 10 minutes from time was cleared badly by Eric Abidal and the ball rebounded off Fernando Llorente and then Gerard Pique before slipping past Valdes.
In the frantic closing minutes Bilbao's Fernando Amorebieta was sent off for a second yellow card and Barca grabbed an equaliser after a free-kick, which pinged around the area until it fell to Messi to score his 14th this season.
"We have shown a lot of character and maintained our unbeaten start," Fabregas told Spanish television. "It was a very difficult match especially in the second half with such a wet pitch.
"It was typical match at San Mames against a team with a great spirit and fantastic fans."
At the Bernabeu, Ronaldo, the world's most expensive player paraded the Golden Boot before kick-off for the 40 league goals he scored last season, and he boosted his bid to win it again with another sparkling performance.
Argentina winger Angel di Maria set up the first three goals for Ronaldo, Pepe and Gonzalo Higuain before being carried off with a torn thigh muscle at the start of the second half.
Ronaldo notched his second goal from the penalty spot in the 54th when Osasuna had defender Eneko Satrustegui sent off, and he completed his fourth league treble this season in the 57th, taking his La Liga tally to 13.
Substitute Karim Benzema wrapped up the scoring with two more goals to complete the rout and Real's 10th successive win in all competitions.
Injury-hit Osasuna, thrashed 8-0 by Barcelona in September, had drawn level at 1-1 in the 31st minute when Ibrahima took advantage of a quickly-taken free-kick to beat Iker Casillas.
"We have totally deserved these 10 victories. They have come from our great effort, without luck, and as a consequence of the good games we have played," Mourinho said.
Valencia are third with 24 points after winning the city derby 2-0 at Levante on Saturday, pushing the season's surprise early pace-setters down to fourth with 23 points.
Granada drew 0-0 at home with fellow strugglers Racing Santander and Villarreal drew 0-0 at Espanyol.
Helder Postiga netted a last-ditch equaliser for Real Zaragoza
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