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Willian Jose sent the hosts ahead in the 11th minute and Juanmi added a second goal in the 34th at a ground where Barcelona had not won in La Liga since 2007.
However Suarez set up Paulinho to pull one back before half-time before scoring twice himself, to ensure Barcelona would remain unbeaten and nine points clear of second-place Atletico Madrid at the top of the table at the halfway stage of the season.
Lionel Messi added some gloss with a perfectly placed free kick for Barcelona's fourth goal in the 85th minute, to end a thrilling game in spectacular fashion.