Messi, 31, has scored eight goals in his past five league games, including a double against Espanyol and a hat trick against Levante.
Messi doubled his side's lead on 53 minutes as he reached another personal milestone. The Argentine received Suarez's lay-off pass before taking a touch to open up some space and firing low into the corner.
And six minutes later, Sergi Roberto took a quick throw-in to play in Suarez, who delicately curled the ball past Asier Riesgo with the outside of his foot.
Barcelona manager Ernesto Valderde praised Messi's performance after the match, saying he had "the feeling that [Messi] will" reach the 500-goal mark.
"But 400 is ridiculous. It's easy to say it, but if you stack [the goals] one after the other ... The numbers are stratospheric, incredible. Messi is an extraordinary player from another galaxy," Valverde said. "And it is not just that he is a goalscorer, it's everything he creates, the goals he gives us, what he generates around him."
Suarez also heaped praise on Messi on yet another momentous night for the Argentine.
"Obviously, we have to be very proud of Messi because every week he demonstrates how good he is and continues to make history with the club," he said.
In total, Messi has now netted 25 goals in just 23 appearances in all competitions this season, including 17 in the league.
Messi has been La Liga's all-time top goal scorer since November 2014, when he overtook Telmo Zarra at the top of the rankings. Former Athletic Bilbao striker Zarra had held the record since the 1950s with 251 goals.
Cristiano Ronaldo has since passed Zarra as well. The Portugal international scored 311 top-flight goals for Real Madrid before leaving for Juventus last summer.
Mexican forward Hugo Sanchez (234 goals) and Madrid legend Raul (228 goals) complete the top five.
Messi has been Barca's top scorer in all competitions for a number of years. After his goal against Eibar, his tally stands at 575, ahead of Cesar (232 goals) and Laszlo Kubala (194 goals).