The Chilean, whose place at the club has been questioned previously, silenced critics with a sublime performance in the absence of Lionel Messi
Alexis Sanchez was the star of the show for Gerardo Martino's side, scoring the opening goal of three after seven minutes before Pedro doubled their advantage shortly afterwards.
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.
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, who have now lost four in a row, were without first-choice goalkeeper Manu Herrera so Tono deputised 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 Alexis to apply a controlled finish.
The hosts were given a scare when a flowing counter-attack resulted in Richmond Boakye crashing a left-foot effort off the post.
However, Barcelona once more cut through Elche's defence after 16 minutes, with Pedro collecting Fabregas' sublime through-ball before rounding the goalkeeper and slotting into an 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 half-time approached and Alexis fired into the side-netting on 43 minutes following a swift breakaway.
Alexis 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 scythed 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 Alexis, 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.