Nigerian Martins gave the visitors the lead in the 40th minute at Municipal de Riazor. Martins slotted home with his left foot after collecting a pass from Míchel.
Los Granotes' Pedro Lopez was dismissed in the 54th minute after handling in the area, a second bookable offence, but Ivan Sanchez missed the resulting penalty.
Home keeper Daniel Aranzubia followed him down the tunnel 16 minutes later and having made all three substitutions, midfielder Alejandro Bergantinos went into the net.
Barkero sealed the points eight minutes from time to move Levante to fourth in the table with a record of six wins, four losses and two draws.
Deportivo are the third from bottom after their sixth defeat of the season.