Vinicius Junior opened the scoring midway through the first half, latching onto Mariano’s flick-on and sending a low finish into the corner.
Elche’s Raul Guti was sent off for a rash challenge on Toni Kroos before Vinicius doubled Real’s lead with a cool finish from Luka Modric’s pass.
Pere Milla scored late on to give Elche hope, but Real held on.
The hosts matched Carlo Ancelotti’s side for large spells of the first half and could have opened the scoring through Lucas Boye, but the Argentine’s shot was superbly saved by Thibaut Courtois.
Lucas Perez should have restored parity after Vinicius’ opener but turned Javier Pastore’s flick well wide of Courtois’ goal.
Real Sociedad can reclaim top spot on Sunday evening when they host Basque rivals Athletic Bilbao.