The Brazilian put the Catalan club ahead after 34 minutes when he fired home following a well worked move and after that the flood gates opened against the bottom side in La Liga.
Six minutes later Luis Suarez slotted past Przemyslaw Tyton after creating the chance with a delightful dummy as he allowed the ball to run through his legs before bearing down on goal.
Lionel Messi, who was the focus of attention following a reported bust-up with Enrique, looked very motivated from the start with some lively runs and converted a penalty just before halftime after Neymar was felled by defender Enzo Roco.
Jordi Alba made it 4-0 with a clinical finish from a Messi pass after 55 minutes and Neymar got his second foal with a deflected drive from distance five minutes later.
Shortly afterwards Neymar was substituted and made it clear he was unhappy as he shook his head when walking off.
It was an important win for Enrique who was criticised for leaving Messi and Neymar on the bench for the defeat by Real Sociedad last Sunday with his team stuttering under him.
The coach put out a first choice strike force against Elche as he could not afford another slip-up and he received a mixed reception from the Camp Nou crowd.
At several times during the game fans chanted his name but they were then drowned out by whistles.
“The only reflection that I will make is that which interests me, and I value the support of the fans to the players. They are the protagonists in this show and I like it that they are supported,” he told a news conference.
“This is basic if we are going to have a good season and it is great if they can back the players.”
It has been a difficult week with presidential elections brought forward a year to the end of this season due to general dissatisfaction with the management of the club.
“I think that our supporters have belief in us and in this sense I am in favour of the decision to hold elections by the president (Josep Maria) Bartomeu as a generous gesture” said Luis Enrique.
“This is a way of bringing some calm and hopefully we can build on that with some good results.”
Barca will face either Real Madrid or Atletico Madrid in the King’s Cup quarter-finals once they finish off Elche. Atletico hold a 2-0 first-leg lead after the home leg on Wednesday.
(Editing by Ken Ferris/Peter Rutherford)