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.
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.
Substitute Karim Benzema wrapped up the scoring with two more goals to complete the rout and Real's 10th successive win in all competitions.
"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.
