Paris Saint-Germain forward Mbappe – who has one league match still to play – scored a double against Dijon on Saturday to close in on Messi, who provided a fine response as Barca's successful LaLiga campaign drew to a close on Sunday.
Barca-owned Marc Cucurella scored against his parent club and although a Messi double had Eibar trailing by the 32nd minute, a glorious Pablo de Blasis strike saw Jasper Cillessen punished for the second time, making it 2-2 just before half-time.
Although both sides crafted chances, the second half was significantly less dramatic as proceedings petered out, bringing the 2018-2019 LaLiga season to an end.
An end-to-end start benefitted Eibar and it took the lead 20 minutes in, Cucurella drilling at Cillessen and the ball squirming underneath the Dutchman.
Messi turned things around just past the half-hour mark, though, equalising with a clinical finish into the bottom-right corner after racing on to Arturo Vidal's pass.
He then produced a delicate chipped finish a minute later after breaking Eibar's offside trap and running through on goal.
Eibar went into the break level thanks to De Blasis' spectacular effort, pouncing on Cillessen's poor headed clearance and catching him out of his goal with a gorgeous long-range volley.
The host looked the more threatening throughout the second period, first going close in the 52nd minute when Sergi Enrich was denied by Cillessen.
Cucurella then wasted a glorious chance nine minutes from time when he blasted over from 10 yards, meaning Eibar had to accept a draw.