Koeman opted for Fati alongside Messi and Philippe Coutinho in his favoured 4-2-3-1 formation, with Antoine Griezmann in the centre-forward role following the sale of Luis Suarez to Atletico Madrid last week.
Messi was playing his first official match since almost leaving Barca in the summer and the Argentine had struck the opening goal under the club’s previous three coaches.
But this time it was Fati who struck first, the 17-year-old blasting a fierce shot into the corner after a cutback from Jordi Alba after 15 minutes.
And three minutes later, the youngster had another as he beat Sergio Asenjo with a low shot after he was set up by Coutinho.
Villarreal had picked up four points in their first two matches under Unai Emery, but the Yellow Submarine could not get going in this one and Messi made it 3-0 later in the half from the penalty spot after Mario Gaspar brought down Fati.
Barcelona were playing their first LaLiga game of the season due to their participation in the Champions League quarter-finals in August and three then became four for the Catalans on the stroke of half-time as Pau Torres turned the ball inadvertently past his own goalkeeper following a Messi ball into the box.
The game dropped in tempo in the second half, but Barcelona remained the better team and came close on a couple of occasions through Messi, who was twice denied by Asenjo.
Miralem Pjanic, Pedri and Francisco Trincao all came on to make their Barca debuts in the closing stages, but there were no further goals in an emphatic win and satisfying start for Koeman at Camp Nou.
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