Suarez's 184th strike for Barcelona saw him move alongside Josep Samitier on the club's all-time goalscoring list.
Victory moved Ernesto Valverde's men to the top of the table, but the Catalan club could be overhauled by Atletico Madrid or Real Madrid by Saturday evening.
— FC Barcelona (@FCBarcelona) October 19, 2019
Eibar made early inroads as they attacked the reigning league champions down the left flank, with Cote having particular joy with his driven crosses.
Cote first caused panic in the Barcelona penalty area when centre-back Clement Lenglet was inches from converting the dangerous low cross into his own net.
The left-back then found Pedro Leon free just inside the area but the forward blazed his effort way over the crossbar.
However, Barca took the lead against the run of play in the 13th minute through Griezmann's composed finish.
Frenchman Lenglet was left unchallenged in his own half and had time to pick his international team-mate out with a searching ball over the Eibar defence.
A slip from right-back Pablo De Blasis gifted Griezmann with time to set himself inside the penalty area before slamming the ball in off the near post for the opening goal.
Eibar continued to threaten from wide areas, with Cote and De Blasis firing in crosses but to no avail, before Messi had a glorious chance to double the visitors' advantage in the 32nd minute.
Barcelona thought they had extended their lead four minutes after the restart when Suarez finished off a fine team move with a deft chip, only to be chalked off for offside.
The visitors started to get in behind Eibar's high-line defence on a regular basis with De Jong dictating play.
And their domination in the second half finally told when Messi grabbed his second goal of the campaign in the 58th minute.
Another slick passing move saw Suarez break into the Eibar area but he looked to have lost control of the ball when he cut back inside, only for Griezmann to instantly slide in Messi who slotted home with an angled drive.
Eibar felt aggrieved as Griezmann, Messi and Suarez combined to wrap up the three points for Barca eight minutes later.
Nelson Semedo appeared to foul Fabian Orellana in the build-up but Barca produced a rapid counter with Griezmann releasing Messi behind the Eibar defence, who unselfishly squared for Suarez to tap into an empty net.
The Uruguay international then rattled a half-volley narrowly wide in the closing stages.