Real Madrid remain two points behind Barca as Zinedine Zidane's dream start to life as boss continued with four goals in the opening 18 minutes of a 5-1 rout of Sporting Gijon.
However, Zidane's joy was also tempered by an injury to Gareth Bale.
Atletico Madrid stay a point clear at the top thanks to Antoine Griezmann's double in a 3-0 victory at Las Palmas.
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Messi paraded his record fifth Ballon d'Or before kick-off and the game at the Camp Nou barely lasted three minutes as a contest.
Suarez was brought down as he rounded Gorka Iraizoz and the Athletic goalkeeper was shown a red card for denying a clear goalscoring opportunity.
Messi converted the resulting spot-kick and Suarez was also heavily involved for the second as he played in Neymar to dink his 19th goal of the season past substitute goalkeeper Iago Herrerin.
Messi wasn't risked for the second-half after feeling a hamstring strain.
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However, the Argentine wasn't missed as brilliant play from Neymar teed up Suarez and Ivan Rakitic to make it 4-0.
Suarez rounded off the scoring with two more to move out in front as La Liga's top-scorer with 18.
"The penalty changed the game and the plan they had to face us, but we knew how to make the most of the circumstances," said Suarez.
Real Madrid's victory was overshadowed by what looked like the recurrence of a calf problem for Bale, which kept him out for a month earlier this season, after he scored his 11th goal in eight games.
Karim Benzema and Cristiano Ronaldo also scored twice each in the first half before the former also picked up a knock in a collision with teammate James Rodriguez.
Madrid have now scored 35 goals in six games at home since losing 4-0 to Barcelona in November.
"We played very well in the first-half. We were concentrated and up for the game from the start. We prepared well and are happy with how the match went," said Zidane.
"Karim just has a knock, which isn't so bad. Gareth is a little more complicated because it is his calf and in his left leg which is a problem.
"He will have a scan tomorrow and we hope it is nothing serious."
Bale headed home Toni Kroos's corner on seven minutes to get the hosts off to the perfect start.
Ronaldo smashed Benzema's through ball into the roof of the net and the latter then acrobatically swivelled to volley home Bale's cross.
Another flowing Madrid move saw Ronaldo sweep home Dani Carvajal's cross at the near post two minutes later.
There was some brief respite for Sporting as Zidane's men waited 23 minutes to add the fifth when brilliant play from Isco freed Benzema to roll in his 20th goal in 19 appearances this season.
Sporting managed to restore some pride when Isma Lopez rounded off a fine team goal just after the hour mark.
Atletico maintained their title challenge as a rock-solid defensive display and Griezmann's goals were again the keys for Diego Simeone's men.
Jan Oblak ensured a 14th clean sheet in 20 games this season with two vital early saves before Filipe Luis fired home his first goal since returning to the club in July to give the visitors a half-time lead.
And with Las Palmas pushing for an equaliser, Koke played in Griezmann twice to finish in the final quarter to take his tally for the season to 18.
Gary Neville's winless streak as Valencia boss in La Liga stretched to six games in a disappointing 2-2 draw at home to relegation threatened Rayo Vallecano.
Valencia twice had to come from behind as Alvaro Negredo's sensational strike from just inside the Rayo half and Paco Alcacer's late equaliser cancelled out goals from Jozabed and Diego Llorente.
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