The trio of star forwards lifted defending champion Barcelona into first place ahead of Real Madrid, which lost 1-0 at Mallorca. Barcelona leads Madrid by one point.
Barcelona enjoyed its best away match of the season.
Griezmann put Barcelona ahead in the 13th minute when Clement Lenglet met his run behind Eibar's high defensive line with a long ball that left him alone to beat goalkeeper Marko Dmitrovic.
Dmitrovic denied Messi before halftime, but couldn't do it again in the 58th after a passing combination by Suárez and Griezmann set up Barcelona's all-time scorer in the area.
The three combined again in the 66th when Griezmann launched Messi with a through ball. Messi then slid the ball over for Suárez to finish off on the break.
Ernesto Valverde's team has won five matches across all competitions after an erratic start to the season.
Barcelona plays at Slavia Prague in the Champions League group phase on Wednesday.