It started with the ball taking on former Hammarby back Joseph's Aidoos arm and Messi made 1-0 on penalty.
Then Celta Vigo's Lucas Olaza went in between with a free kick goal before Lionel Messi answered with two.
Celta's goalkeeper Ruben Blanco saw two almost identical free kicks screw in to his left.
Midfielder Sergio Busquets then set the score to 4-1.
With his hattrick Messi rival Cristiano Ronaldo's La Liga record as both have now scored three goals in a match 34 times.