Luis Enrique's side enjoyed the perfect preparation for their Champions League semi-final clash with Bayern Munich with a thumping win on Saturday
The game was goalless until Ivan Rakitic's 37th-minute opener, which saw Cordoba's resilience quickly crumble, as Suarez made it 2-0 to the visitors soon after.
Messi added his 50th goal of the season across all competitions just after the restart and Suarez doubled his tally seven minutes later, before Gerard Pique headed in the fifth with 63 minutes played.
Messi netted his second with a tap-in 10 minutes from time, before Neymar and Suarez rounded off another stunning performance from Barca, with the Uruguayan clinching his hat-trick and relegating Cordoba in humiliating fashion.
Despite the fact Bayern Munich await in the Champions League semi-finals next week, Luis Enrique surprisingly opted against resting key players and named arguably his strongest starting XI.
The away side were in complete control of proceedings right from the start, though Cordoba initially looked resolute at the back, keeping Messi, Suarez and Neymar at bay.
Cordoba's defence began to show signs of frailty in the 27th minute, as Messi weaved through in stunning style, skipping past four challenges before dragging a tame finish straight at Juan Carlos in the hosts' goal.
But Barca's patience finally paid off soon after.
Once again Messi was the architect, clipping a delightful delivery into the left side of the area for Rakitic, who brought the ball down and smashed an unstoppable half-volley into the roof of the net.
And their second followed swiftly on the stroke of half-time, Suarez racing on to Andres Iniesta's cross and guiding the ball beyond the helpless Juan Carlos.
Messi eventually got his name on the scoresheet in the first minute of the second half, arriving late in the penalty area before guiding a precise header into the bottom-left corner.
Cordoba's heads dropped and Barca continued to pile forward ruthlessly, opening up a four-goal lead in the 53rd minute, as Suarez glanced Dani Alves' right-wing cross in off the post with his head.
The visitors were completely dominant and added their fifth just after the hour mark, Pique rising highest to meet Rakitic's corner and nod in.
After a 15-minute respite for Cordoba, Messi tapped in following good work from Suarez, while Neymar netted an 85th-minute penalty after he had been hauled down.
There was still time for Suarez to add to an already embarrassing day for Cordoba, the Uruguay star knocking into an open net after pouncing on a blocked shot two minutes from time, wrapping things up and sending a warning shot to Bayern.