The on-form Borja Baston fired Eibar into a shock lead just 10 minutes in, but Suarez restored parity with a bullet header and then put Barca in front just after the break.
Eibar were handed a lifeline when Javier Mascherano saw a straight red card for protesting with the referee seven minutes from time.
But Suarez promptly sealed his hat-trick and a vital three points with another fine finish from Neymar's pass.
Victory takes Barca level with Real Madrid on 21 points, but Los Blancos remain on top thanks to their superior goal difference.
There was a moment of reflection before kick-off as the Barcelona players entered the field wearing orange t-shirts to honour legendary former Barca player and coach Johan Cruyff, who was diagnosed with lung cancer this week.
And the Camp Nou was stunned early on as Eibar went in front when on-loan Atletico Madrid striker Baston swept home his sixth goal of the season after Claudio Bravo had blocked Keko's initial attempt.
Barca youngster Sandro Ramirez has struggled with the unenviable task of standing in for Messi during the four-time World Player of the Year's absence.
However, the 20-year-old had a major role to play in a controversial equaliser 11 minutes later as he brilliantly volleyed Sergio Busquets's ball over the top back across goal for Suarez to power home a header from close range.
Yet, replays showed the winger had been in an offside position when he latched onto Busquets's pass.
Despite Barca's dominance of possession, Eibar went on to have the better chances before half-time.
Simone Verdi sent a rising drive into the side-netting before Keko had a goalbound effort cleared off the line by Marc Bartra.
The visitors had the first chance after the break too when Baston's weak header was easily claimed by Bravo.
And Eibar's profligacy was punished seconds later when Neymar weaved past two defenders and squared for Suarez to thump his second goal of the evening into the bottom corner.
Mascherano could be in major trouble as his straight red may lead to a four-game ban that would see him miss the clash with Real Madrid on November 21.
However, Suarez ensured Barca weren't to lose any ground on Rafael Benitez's men in the meantime with a brilliant finish after controlling Neymar's pass on his chest two minutes later.
Elsewhere, the pressure on former Manchester United manager David Moyes was eased as his Real Sociedad side thrashed Levante 4-0 on Sunday to move three points clear of the relegation zone.
Former Arsenal forward Carlos Vela opened and rounded off the scoring either side of goals from Imanol Agirretxe and Asier Illaramendi.
Villarreal's dip in form continued as they moved up to fourth despite a disappointing 0-0 draw at Las Palmas.
However, Atletico Madrid can overtake the Yellow Submarine and move up to third with victory over Valencia later on Sunday.
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