Liverpool gave their hopes of reaching the Champions League knock-out phase a massive boost with a superb victory.
Jorge Andrade's own goal after 13 minutes gave Liverpool coach Rafael Benitez a winning return to Spain.
Andrade turned in John Arne Riise's cross under pressure from Milan Baros and Liverpool's margin of victory should have been greater.
Baros and Riise were denied by keeper Jose Molina, and the Norwegian also had a shot kicked off the line by Andrade.
Liverpool suffered a blow when Xabi Alonso was ruled out with a calf injury, with Igor Biscan coming into a central midfield role.
It barely disturbed Liverpool's rhythm as they dominated an impressive opening 45 minutes when they could have opened up an unassailable lead.
Baros should have put Liverpool ahead inside the first 30 seconds as he raced clear, but Deportivo goalkeeper Molina plunged bravely at his feet.
Deportivo responded with a shot from Sergio that flew narrowly wide before Liverpool deservedly went ahead in the 13th minute.
Biscan broke from midfield to set up Riise, whose cross was turned into his own net by Andrade under pressure from Baros.
And four minutes before the interval, Liverpool were twice within a whisker of doubling their advantage.
Harry Kewell set up Riise, whose first effort was blocked by Molina, before Andrade swept another effort from the Norwegian off the line from the rebound.
Deportivo made a half-time change, sending on Pablo Amo for Cesar, and Alberto Luque quickly tested Chris Kirkland with a free-kick from 20 yards.
And Amo was almost an instant hero, heading wide from a yard out in an incident which saw Kirkland injured in a clash with his own player Biscan.
Liverpool made a change after 57 minutes, sending on Steve Finnan for the tiring Kewell.
Deportivo enjoyed plenty of possession, but Liverpool defended magnificently and were well worthy of the win.
D Coruna: Molina, Romero, Cesar (Pablo Amo 45), Andrade,
Hector (Scaloni 65), Sergio, Valeron, Duscher, Victor,
Pandiani (Tristan 59), Luque.
Subs Not Used: Munua, Fran, Munitis, Capdevila.
Booked: Cesar, Andrade.
Liverpool: Kirkland, Josemi, Hyypia, Carragher, Traore, Riise,
Luis Garcia (Alonso 91), Hamann, Biscan, Kewell (Finnan 58),
Baros (Sinama Pongolle 84).
Subs Not Used: Dudek, Henchoz, Diao, Warnock.
Booked: Carragher, Biscan, Hyypia.
Goals: Andrade 14 og.
Ref: Wolfgang Stark (Germany).