Xavi's side have moved a point clear of second-placed Real Madrid at the summit, with the reigning champions due to face Atletico Madrid in the Madrid derby on Sunday night.
Barcelona had a man advantage from the 14th minute, with Gonzalo Verdu shown a red card for fouling Lewandowski when the Poland international was through on goal.
Memphis Depay - handed a rare start - and Lewandowski both came close to scoring just before the 30-minute mark, but the former saw his effort saved by Elche goalkeeper Edgar Badia, before Lewandowski just missed the target in a separate chance two minutes later.
Xavi's side had the lead in the 34th minute, though, with Lewandowski on hand to convert a cross from Alejandro Balde, who again had an impressive game at left-back with two assists.
The hosts had a second before the interval, with Memphis finishing high into the back of the net after more smart work for Balde.
Elche coach Francisco Rodriguez was also sent off late in the first period after Pedri had a goal disallowed for offside, capping a miserable opening 45 minutes for the visitors.
Barcelona's third came shortly after the break, with Lewandowski having the simple task of finishing after smart work from Memphis, who made a strong impression on the afternoon.
Memphis should have scored a second just before the hour when he met a cross from Ousmane Dembele, but the Dutchman somehow sent his header wide of the post.
Raphinha and Ronald Araujo were among the players to have chances of their own in the latter stages, but Elche managed to keep the score down to 3-0, with the hosts moving to the summit and the visitors remaining bottom of the division with just one point to show from six matches.