After a bright opening, Messi broke the deadlock with fierce free-kick in the 22nd minute.
His second came 10 minutes after half-time with a cool finish from inside the area, before Gerard Pique put gloss on the scoreline with a header from Neymar’s cross and Messi completed his treble when he feigned his shot before guiding an effort into the corner.
Kolbeinn Sigthorsson could have grabbed a consolation, but he saw his 77th-minute penalty saved by Victor Valdes.
Martino included Messi and Neymar in Barcelona’s starting XI, with the latter making his Champions League debut.
The Dutch side had Siem de Jong available, following his recovery from a collapsed lung.
Barcelona dominated possession in the opening exchanges and they took the lead when Messi once again displayed his brilliance with a stunning free kick, 22 minutes in.
After being felled by Lerin Duarte on the right, he picked himself up to hit a thunderous strike, with the ball going in off the the post as Kenneth Vermeer failed to keep it out at full stretch.
Ajax nearly stole an equaliser on the half-hour mark when Bojan curled a cross deep towards the far post and a diving header from Ricardo van Rhijn forced a smart reaction save from Valdes.
The visitors could have had a penalty when Sigthorsson went to ground under a challenge from Pique. The loose ball was met by Duarte, but he placed his first-time effort straight at Valdes.
After surviving those scares, Messi effectively ended the game as a contest in the 55th minute.
He controlled a Sergio Busquets pass on the right side of the box before feinting inside Stefano Denswil and firing his effort past Vermeer.
Barcelona continued to press and Neymar and Dani Alves both had efforts blocked.
And the pressure told in the 69th minute when Neymar delivered an in-swinging cross from the left that was met by the head of Pique to make it 3-0.
Messi then completed his hat-trick with 15 minutes remaining.
