The €94 million striker finally netted in the Primera Division, while Lionel Messi struck twice, with Pedro and Gerard Pique bagging one goal apiece
Pedro broke the deadlock with just three minutes on the clock and the winger then turned provided for Suarez, who opened his Primera Division account at the eighth attempt with an instinctive finish shortly after half-time.
Gerard Pique made it 3-0 by heading home a Xavi free kick and before Lionel Messi put the seal on an ultimately facile victory with two late right-footed strikes.
Luis Enrique opted to leave Neymar on the bench despite the Brazilian overcoming an ankle injury, with Pedro's Copa del Rey hat-trick against Huesca earning the Spain international a start in La Liga.
And the coach's decision was vindicated with just over a minute played, as Rakitic's lofted pass picked out Pedro's darting run and the forward coolly steered past Juan Carlos.
Barcelona continued to dictate the game and went agonisingly close to doubling their lead in the 12th minute, but Suarez's acrobatic volley from a Jordi Alba cross found the side-netting.
Cordoba eventually tested Claudio Bravo with 26 minutes gone, as Borja Garcia tried his luck from the left flank and almost saw his attempt creep inside the Chilean goalkeeper's right-hand post.
But that only proved to be a brief moment of respite for the visitors, who had another fortunate escape 10 minutes later as Pedro fired just off target from the edge of the area, before Pique saw a header saved on the stroke of half-time.
It did not take long for Barca to extend their lead in the second half, though, as Pedro drove into the left side of the area in the 53rd minute and cut the ball back for Suarez, who prodded the ball under Juan Carlos.
Barca almost scored again just a few moments later, but midfielder Rakitic's venomous long-range drive rattled the post.
Juan Carlos was beaten again 10 minutes from time, though, as Pique headed Xavi's corner past the helpless goalkeeper and Messi, who had been quite by his high standards, grabbed a fourth with a tidy finish into the top-left corner after another set-piece was not cleared.
Messi completed the rout in stoppage time, brilliantly bringing down Alba's cross before slamming beyond Juan Carlos with a clinical finish to take his tally to 15 La Liga goals for the season and end another prolific year in style.