Eibar surprised early at the Camp Nou with a goal by striker Borja Baston in the 10th minute, but Suarez helped the defending champions rally with goals in the 21st, 48th and 85th minutes.
Barcelona played the final minutes with 10 men after defender Javier Mascherano was sent off with a straight red card.
Barcelona and Real Madrid have 21 points each after nine rounds, but Madrid remained ahead on goal difference after defeating fourth-place Celta Vigo 3-1 on Saturday in Vigo.
Celta fell three points behind the leaders.
Atletico Madrid moved to third by beating Valencia 2-1 on Sunday with goals by Jackson Martinez in the 32nd and Yannick Carrasco in the 40th at the Vicente Calderon Stadium in Madrid.
Paco Alcacer scored Valencia's goal by converting a penalty kick in the 72nd.
Barcelona struggled to take control of the match at the Camp Nou and it was Eibar which created many of the dangerous scoring opportunities.
The Basque club took the lead after a quick counterattack as Baston scored with a shot from outside the area that Barcelona goalkeeper Claudio Bravo couldn't keep out.
Suarez equalized with a header from close range after a pass by Sandro, who appeared to be offside when he received a long cross inside the area.
The Uruguay striker added to the lead with a low shot from near the penalty spot after a nice run by Neymar, and closed the scoring with another low strike from inside the area after another set up by Neymar.
"We've won a very complicated match and I'm glad that I could help with important goals," Suarez said.
On the one-year anniversary of his first game with Barcelona, Suarez took his tally to 10 goals this season. He has seven in the league, only one less than Neymar.
"We know the importance of Luis," Barcelona coach Luis Enrique said. "What he gives you is invaluable."
Barcelona, still without the injured Lionel Messi, had lost two of its last four league games. It also had to rally from a goal down in its previous home match, a 5-2 win over Rayo Vallecano last week.
Eibar was unbeaten in four matches entering the matchup against the defending champions.
It dropped to seventh place in the 20-team standings following the loss.
Mascherano was sent off in the 83rd after complaining to the officials and could face a long suspension if found guilty of insulting the linesman.
Earlier Sunday, fifth-place Villarreal missed a chance to stay near the leaders after a 0-0 draw at newcomer Las Palmas, while Real Sociedad defeated host Levante 4-0 with a pair of goals by forward Carlos Vela.