Barca were missing both star playmakers Lionel Messi and Neymar - Messi is recovering from injury and Neymar started on the bench. But they still boasted far too much firepower for lowly Elche, with Pedro adding to Alexis' first treble for the Catalan giants.
The hosts also had the luxury of missing a second-half penalty from Xavi as they reclaimed top spot from Atletico Madrid, who went three points clear with their win at Malaga on Saturday.
Even though he was still without Messi, Barca boss Gerardo Martino opted to rest Neymar from his starting line-up but there was little chance of the move backfiring as the reigning champions struck twice inside the opening 16 minutes.
Alexis notched the first after seven minutes when he side-footed a Jordi Alba cross into the roof of the net and Pedro doubled the lead when he latched on to a superb through-ball from Cesc Fabregas, rounded Elche goalkeeper Antonio Tono and rolled his shot into an empty net.
Barca did survive a brief scare in between those two goals when Elche striker Richmond Boakye saw his left-footed shot crash back off the near post, but otherwise the home side looked untroubled as they made it nine league wins from nine at home this season.
Pedro twice tried his luck without success as Barca sought a third goal, which really should have arrived two minutes into the second half when they won penalty after Fabregas had been sent flying inside the area by the sliding Cristian Sapunaru.
Xavi took the penalty but he side-footed his shot wide of the right-hand post.
Barca need not have worried, though, as they did make it 3-0 in the 63rd minute when Fabregas and Pedro combined to set up Alexis for a straightforward tap-in from seven yards out.
Alexis will not get many easier finishes than that this season, but he had to produce something a little more special for his hat-trick goal.
A foul by Alberto Botia on the Chile international 25 yards out saw Barca awarded a free-kick, and Alexis stepped up to curl a brilliant effort into the top corner of the net for his 11th league goal of the season.
Neymar was finally unleashed with 10 minutes to go, replacing Pedro, but the points were already in the bag by then as Elche slumped to their fourth successive loss.