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The hosts opened the scoring after just seven minutes courtesy of Neymar, who tapped in from close range after receiving an unselfish pass from Messi.
Neymar came close to doubling his tally moments later after once more being picked out by Messi, though the Brazilian failed to find the net from a tight angle.
Celta had a chance to level the contest through Augusto Fernandez, who found space in the box following a corner, but his effort was just wide of the post.
Messi added a second for Barca with 31 minutes on the clock, driving into the box before rounding Yoel and finishing into the bottom corner to give his side some breathing space.
Ten minutes into the second half the Sky Blues had another opportunity to open their account for the night. Fernandez was again involved, meeting a cross from Levy Madina and forcing substitute goalkeeper Jose Pinto into a fine close-range save.
Luis Enrique's side would live to rue the missed opportunity, as with 68 minutes gone Neymar added to his tally when he collected a long pass from Alexis Sanchez, smashing the ball into the bottom corner.