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The hosts were quickest off the starting blocks and combined to slice through the opposition defence on numerous occasions in the first few minutes, but the visitors managed to hold tight at the back.
Barca's pressure duly paid off when Neymar opened the scoring in the 35th minute, Luis Suarez's ball across the face of Przemyslaw Tyton's goalmouth falling to the Brazilian, who had the simple task of prodding home.
Just five minutes later, the Uruguayan striker doubled his side's lead when he surged from the left flank towards goal, before drilling a low shot past the Elche keeper.
Lionel Messi then had a goal disallowed for offside, but Barca nevertheless notched up a third strike when Neymar was brought down by Enzo Roco in first-half injury time, and the Argentine prodigy duly converted from the spot.
The hosts picked up from where they left off in the second half and scored a fourth in the 56th minute, Messi slotting through to Jordi Alba, who took a touch before firing past Tyton.
On the hour, Neymar waltzed through the beleaguered Elche defence before letting rip with a venomous shot that bounced off the underside of the crossbar and into the net.
Barcelona continued to probe forward in a bid to add further gloss to the scoreline, as Messi and Pedro notched up further chances on goal, but the visitors were able to stem the flow until the final whistle.
Barring a miracle at the Estadio Manuel Martinez Valero in the second leg, Luis Enrique's charges will be in the quarter-finals of the Copa del Rey.