The attacking duo combined to ruthless effect after Gerard Pique's early goal against the home side, which played much of the second half with 10 men.
Barca had already beaten Elche three times this season — and scored 12 goals in the process — ahead of the La Liga clash at Estadio Manuel Martinez Valero and the Catalan giants heaped more misery on Fran Escriba's struggling side.
Gerard Pique marked his 200th La Liga appearance with the opening goal in the first half and Barca ran riot in the second, with Messi scoring from the penalty spot before Elche had Faycal Fajr sent off for a second bookable offense.
The majestic Messi then laid on two goals for Neymar, who struck twice in as many minutes to take his goal tally for the season to 19, and Messi added a fifth with his 31st of another outstanding campaign before Pedro got in on the act in stoppage time.
Luis Enrique's side is just a point behind first-place Real Madrid, which won 2-1 at Cordoba earlier in the day, while defeat for Elche came after back-to-back La Liga wins. Elche remains mired in trouble just above the relegation zone.
Luis Suarez was dropped to the bench on his 28th birthday despite this week being backed to put a disappointing start to his Barca career behind him by Luis Enrique.
Xavi was restored to the starting lineup and the fit-again midfielder almost created the opening goal when he picked out Jordi Alba with a clever free kick 13 minutes in, but the left back's well-struck volley flashed over the crossbar.
Elche was working hard off the ball to frustrate Barca, but Messi came close with a free kick that flew narrowly wide.
The home side posed an attacking threat of its own and Fajr called Claudio Bravo into action following a swift counterattack and then worked the Barca goalkeeper again with a long-range free kick after 27 minutes.
Neymar then wasted a glorious chance to open the scoring, when he miskicked with only Przemyslaw Tyton to beat after Messi breached the Elche defense with a lovely ball over the top.
Barca was in front 10 minutes before the break, though, when Pique showed great composure to bring down Xavi's free kick with his chest before beating Tyton with a deflected right-footed finish.
Another sublime pass from Messi opened up the Elche defense once again seven minutes after the interval, but Tyton was alert to the danger as he came out to thwart Pedro.
Barca strengthened its grip on the game a minute later, though, when Elche were caught on the break and Sergio Pelegrin was ruled to have upended Neymar when the Brazil forward jinked past him.
Messi casually slotted home the resulting spot kick to take his La Liga tally for the season to 20 and there was worse to come for the home side as Fajr was shown a second yellow card for a late challenge on Rafinha only 12 minutes into the second half.
Neymar added to Elche's misery by adding a third goal with a clinical right-footed finish after being set up by a perfectly weighted pass from the outstanding Messi after 69 minutes.
And the pair combined once again two minutes later, with Neymar again applying the finish after playing a one-two with the imperious Argentina forward.
Messi then doubled his tally, as he created space for himself in the area and fired into the far corner with his left foot two minutes from time.
Neymar laid on a sixth goal for Pedro, who rounded off the scoring when he tapped home following a rapid counterattack in stoppage time.