Atletico have 27 points from 15 matches, 10 off leaders Barcelona and eight behind Real Madrid having played a game more than their two rivals. Bottom side Elche have four points and are eight adrift of the safety zone.
Atletico struggled to control the game and create any clear chances in the opening period before Elche defender Gonzalu Verdu was shown a red card one minute into first-half stoppage time after dragging Morata down just outside the box.
Atletico defender Mario Hermoso then received his marching orders in the 53rd minute at the Metropolitano Stadium after picking up two bookings in three minutes for fouls.
Three minutes later Felix headed in the opening goal after Antoine Griezmann found him inside the box.
Morata doubled the lead in the 74th with a shot that was deflected over goalkeeper Edgar Badia.
Elche were reduced to nine men in the 90th when Domingos Quina received his second booking for a challenge on Morata.