The 17-year-old, who was officially promoted to Barca's first-team squad last week, added a second in the 19th minute after Coutinho led a counter-attack.
Messi added a third on 35 minutes, scoring a penalty after Fati was fouled by Mario Gaspar. Villarreal goalkeeper Sergio Asenjo got a hand to the spot-kick but was unable to stop it.
The hosts continued to dominate the game in the second half without managing to add to their hefty advantage.
Koeman, who succeeded Quique Setien in August after the crushing 8-2 defeat to Bayern Munich in the Champions League, could hardly have hoped for a better start after the chaotic close season overshadowed by Messi's attempt to leave the club.
He had unleashed his latest attack on the club's hierarchy only on Friday, criticising the manner in which his former strike partner Luis Suarez had been forced out of the club.
Suarez enjoyed a superb debut at Atletico Madrid earlier on Sunday but here Barca did not seem to be missing him as they played with more pace and freedom than for many parts of last season, launching quickfire counters as well as building from the back.
"It was a good game for us especially in the first half, we were much better than them, we created lots of chances and scored four goals," Barca midfielder Sergio Busquets said.
"We dropped our intensity a bit in the second half and they made things more difficult for us but overall we leave our first game with good sensations. We're close to the team we want to be."
Ousmane Dembele also came on for his first competitive appearance for Barca since last November after returning from hamstring surgery.
Information from Reuters was used in this report.