The Chilean bagged three goals while Pedro added another as the Catalans rolled to a comfortable triumph despite once again missing Lionel Messi.
Alexis Sanchez was the star of the show for Gerardo Martino's side, scoring the opening goal after seven minutes before Pedro doubled the advantage shortly after. The Chile international was to complete a hat trick with two goals after the break, though only after Xavi had missed a penalty.
Elche had no answer as Gerardo Martino's men began 2014 in style with their 16th win in 18 matches, taking their tally of goals in La Liga this season to 53 in the process.
Barcelona made a great start with Sanchez's neat side-foot finish after seven minutes, and the lead was doubled just after the quarter-hour mark by a Pedro tap-in.
Fabregas played a part in both of those goals and earned Barcelona a penalty shortly after the restart, which Xavi put wide.
The missed spot kick did not have any affect on the result as Sanchez tapped home Pedro's center in the 63rd minute, before rounding off the win with a glorious free kick.
Victor Valdes returned in goal for the hosts after being sidelined since November with a calf injury, while Neymar was only named on the bench after returning from suspension. Elche, which has now lost four in a row, was without first-choice goalkeeper Manu Herrera, so Tono deputized between the sticks.
Barcelona started well and opened the scoring when Jordi Alba collected Fabregas' cross-field pass and whipped the ball into the box for Sanchez to apply a controlled finish.
The hosts were given a scare when a flowing counterattack resulted in Richmond Boakye crashing a left-footed effort off the post.
However, Barcelona once more cut through Elche's defense after 16 minutes, with Pedro collecting Fabregas' sublime through ball before rounding the goalkeeper and slotting into the empty net.
Pedro continued to be a menace and twice came close to doubling his tally, curling wide from Fabregas' pass across the box, before lofting an inventive half-volley wide. Barcelona retained control of proceedings as halftime approached and Sanchez fired into the side-netting in the 43rd minute following a swift breakaway.
Sanchez then saw a close-range header deflect wide at the start of the second half, and Xavi put a penalty wide of Tono's right-hand post in the 48th minute after Fabregas had been taken down by Cristian Sapunaru.
Xavi's miss failed to derail Barcelona and the champions made it 3-0 when Pedro broke down the left and laid invitingly into the path of Sanchez, who tapped home.
The Chile international completed the rout with a sublime curling free kick from 25 yards six minutes later.
Barcelona slowed the tempo in the closing stages but had legitimate claims for a second penalty when Andres Iniesta appeared to be upended by Alberto Botia.