Suarez bagged an early brace before Lionel Messi scored twice either side of the interval to take his career tally to 500 goals. Neymar missed a penalty in a one-sided first-half but his Argentine counterpart completed his hat-trick late.
The Uruguayan then joined his striking partner by heading in his third before claiming a fourth in the closing stages as the cup holders heaped more pressure on Los Che manager Gary Neville.
Neville’s side were looking to put their league woes behind them, having failed to register a win in eight La Liga games since the current England coach was appointed in December.
The Copa del Rey has been a happy hunting ground for Valencia, however, as they reached the last four whereas Barcelona were seeking to extend their 26-game unbeaten streak in all competitions.
But the visitors got off the worst possible start after seven minutes.
Neymar dispossessed Andre Gomes inside the Valencia half before playing through Suarez, who fired a low shot past Mathew Ryan in the Valencia goal.
Barca were on the front foot once again as Neymar picked out Messi, who set up Suarez – but Jose Gaya was able to block the former Liverpool man’s shot.
But Suarez made it 33 goals in 33 appearances moments later as Sergio Busquets floated in a beautifully weighted pass into the path of Alex Vidal, whose first-time pass was blasted past Ryan by the Uruguayan.
Los Che struggled and things got worst for Neville’s side as the hosts grabbed a third goal on the half an hour mark.
Andres Iniesta found Neymar on the edge of the area before the Brazilian’s cheeky backheel eventually found the surging run of Messi, who slotted home.
Daniel Parejo headed over as the visitors had a rare glimpse at goal but they were lucky not to concede a fourth in the closing minutes of the first half.
Messi smacked his effort off the crossbar before Shkodran Mustafi was given his marching orders after denying a goalscoring opportunity for his challenge on the Argentinian inside the box.
And with virtually the last kick on an eventful opening period Neymar missed the resulting penalty as his shot hit the post.
The home fans roared on the introduction of Denis Cheryshev, whose appearance in an earlier round for rivals Real Madrid got them eliminated from the competition because of suspension, early in the second half.
Pique, Messi and Busquets had chances to stretch the lead further after the break before the Blaugrana fans were on their feet again as the Argentinian superstar grabbed his second.
Suarez turned provider as his flick met the feet of Messi, who shuffled past the hapless Valencia defence before firing into the bottom corner - his 500th career goal for Barcelona.
Valencia thought they had scored a consolation in the 67th minute but Rodrigo’s effort was ruled out for offside.
Things went from bad to worst as Parejo carelessly lost possession before Messi beat Ryan at the near post.
Barca continued to run riot and they made it 6-0 as Suarez completed his hat-trick when Sergi Roberto found the forward at the far post to head in.
The former Liverpool striker increased the visitors’ misery as his 35th goal of the season, with next week’s second leg at the Mesalla a virtual formality for Luis Enrique’s men.