The Chilean was ably assisted by superb performances from Pedro and Cesc Fabregas on Sunday, the former notching his sixth goal in as many league matches and continuing to shine in the absence of Argentina international Lionel Messi.
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-footed finish after seven minutes, and the lead was doubled just after the quarter-of-an-hour mark by a Pedro tap-in.
The missed spot-kick did not have any affect on the result as Sanchez tapped home Pedro's centre on 63 minutes, before rounding off the win with a glorious free-kick.
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.
However, Barcelona once more cut through Elche's defence after 16 minutes, with Pedro collecting Fabregas' sublime throughball 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.
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.
