The South American duo did the damage in the first half with two fine goals, before Antoine Griezmann gave his side a glimmer of hope after the break.
Barcelona took a big step towards another Copa del Rey final with a 2-1 first-leg victory against Atletico Madrid at the Vicente Calderon.
First-half strikes from Lionel Messi and Luis Suarez secured the victory that puts the holders in control of the semi-final ahead of the return match at Camp Nou next Tuesday.
Diego Simeone's side have had success against Barca in recent years with a devastating counterattacking style, but they were on the receiving end of the same tactic early in the first half and were punished by a brilliant run and finish from Suarez with only seven minutes played.
Messi made it 2-0 with a ferocious strike from the edge of the penalty area to silence the Calderon and leave Atletico facing a mountain to climb.
The 2013 winners did show some improvement after the break, however, and Antoine Griezmann halved the deficit just shy the hour mark before missing a clear chance to level the scores moments later.
But Barca - who almost snatched a third through a Messi free-kick - held firm in the closing stages to see out a crucial win that will make them firm favourites to reach a fourth final in as many seasons.
Luis Enrique - still unable to call on Sergio Busquets or Andres Iniesta - named a robust midfield of Javier Mascherano, Ivan Rakitic and Andre Gomes. And it was a Mascherano challenge on halfway that allowed Suarez to open the scoring.
The Argentine's sliding challenge sent the ball into Suarez's path and the striker drove at the heart of the Atleti defence, slipped the ball around Stefan Savic and prodded low past Miguel Angel Moya and into the corner of the net.
Atleti responded well, but Barca, content to sit back and hit on the break, almost got a second midway through the half when Neymar cut into the box and shot straight at Moya.
Neymar and Suarez nearly combined to good effect in the penalty area, but Messi followed a more direct route to double the lead 12 minutes before half-time, blasting a left-foot shot in off Moya's right-hand post from the edge of the area.
Koke arrowed a shot just over four minutes before half-time and the Spain midfielder combined superbly with substitute Fernando Torres to move clean through on goal moments after the restart, only for Jordi Alba to make a crucial covering block just as he pulled the trigger.
Griezmann had been starved of service for much of the first hour, but he handed the hosts a lifeline in the tie as Atleti punished some slack Barca defending with a route one set-piece.
Koke flighted a free-kick from near the centre circle to the left side of the penalty area and after Diego Godin - who was completely unmarked - headed the ball across goal, Griezmann rose highest to nod it into the net from six yards.
Jasper Cillessen stood up well to block what looked a certain second for Griezmann, who shot straight at the Netherlands goalkeeper following Filipe Luis' cut-back, as Atleti began to attack in earnest with the Vicente Calderon roaring its approval.
Moya clawed a dipping Messi free-kick away from the top corner and Neymar blasted a presentable chance over the crossbar as Atleti committed men into the attack, before the Brazilian fired another set-piece over Moya's goal from close range.
Simeone urged his players forward in the final few minutes and Torres twice shot inches wide of the post when picked out in space in the area, but there was to be no late leveller for Atleti, who face a huge task to overturn the aggregate deficit when they visit Camp Nou next week.
