Barcelona opened up a five-point gap over Real Madrid before the second-place side visits Sevilla later, while Cordoba's 17th round without a win confirmed the team's relegation.
Messi played Ivan Rakitic clear to start the rout in the 42nd minute, and Suarez got his first goal in first-half injury time.
Dani Alves crossed for Messi and Suarez to add two headed goals before Gerard Pique netted a header of this own.
Messi, Neymar and Suarez then polished off the demolition.
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