Cristiano Ronaldo hit a record-breaking 25th La Liga treble as Real Madrid won a tricky match at Sevilla 3-2 on Saturday.
Barca moved a step closer to their fifth La Liga title in seven years but Real trail them by only two points with three games left in the season.
Champions Atletico Madrid lacked creativity against Athletic Bilbao and were held to a 0-0 draw but remained in the driving seat for third place with a seven-point lead over Valencia who play Eibar on Sunday.
Suarez's prolific form has brought the Uruguayan 17 goals from his last 16 games in all competitions while Messi has 40 goals -- two less than top scorer Ronaldo.
Ronaldo now has the highest number of hat-tricks in La Liga history, one more than Messi.
He headed home an Isco cross after 36 minutes and a minute later Javier Hernandez flicked on a James Rodriguez cross to the Portuguese who beat keeper Sergio Rico from six yards.
Substitute Gareth Bale crossed for a fine header by Ronaldo from a narrow angle to complete his hat-trick after 69 minutes and condemn Sevilla to their first home defeat in 35 matches in all competitions.
Ivan Rakitic volleyed into the roof of the net to give Barca the lead after 42 minutes at Cordoba and Suarez slotted home in first-half stoppage time.
Messi headed in straight after the restart, with further headers from Suarez and Gerard Pique as Barcelona hit their stride.
Messi scored into an open net with 10 minutes to go and Neymar converted a penalty after he was brought down by defender Adrian Gunino.
Barca showed no mercy with Suarez striking again two minutes from time to seal a resounding win.
"Football is a very complex sport. We are on the right road but you have to start again before each game and look to create the play in order to get the results," Barca coach Luis Enrique told reporters.
"The goals from Ivan and Luis have got us going and then in the second half we were very effective in front of goal.
"These were three key points and I am very pleased with the result."
Real were still without Luka Modric and Karim Benzema while Bale was only fit to start the game on the bench.
Sevilla, who held Barcelona to a draw last month, started well and caused problems for the Real defence down the wings with Jose Antonio Reyes in particular a threat.
The Andalusian side though were down to 10 men when Ronaldo pounced for his first two goals as Grzegorz Krychowiak was on the sidelines after a clash of heads with Sergio Ramos.
Sevilla fought back and Carlos Bacca scored a penalty before halftime after Ramos felled Aleix Vidal.
After Ronaldo added his third Vicente Iborra scored from a Vidal cross and although Real had to resort to last-ditch defending they hung on for the victory.
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