Liverpool's coach must have wanted this victory so much on his first visit back home since quitting Valencia for Merseyside.
It came from a Jorge Andrade first-half own goal but was fashioned by a gritty performance from Liverpool, who had unlikely heroes like Igor Biscan to thank for the vital Group A points.
And with a side hit by injuries, Benitez saw his men fight for every yard in the second-half.
But with Olympiacos beating group leaders Monaco, it means if Liverpool win their next match in Monaco later this month, they will be assured a place in the knock-out stage.
Liverpool, already without broken leg victim Djibril Cisse, received a blow before the start when midfielder Xabi Alonso was ruled out with a calf injury which had troubled him since Saturday's draw at Blackburn.
Under such circumstances, with Liverpool already stripped of Steven Gerrard and Cisse, a backs-against-the-wall approach would have been forgiven although chances which came their way would not have to be wasted.
But that was what happened within 30 seconds of the kick-off when Dietmar Hamann and
Garcia created a chance for Milan Baros through the middle.
The Czech was clean away but tried to go round Jose Molina and the 'keeper stole the ball when the forward should have buried the opportunity.
But after 14 minutes Liverpool got the ball in the net. Biscan ran through midfield and fed John Arne Riise on the left.
The Norwegian's cross arrowed into the box and Andrade under pressure from Baros turned the ball into his own net.
Liverpool aimed to hit on the break and produced movement and passing which continually threatened to upset the Galician side, with Harry Kewell dropping deep to link with the astute Baros.
Andrade and Cesar struggled to contain him, with the Czech star enjoying his lone striker role. On 37 minutes Cesar Martin was booked for bringing down Baros on the left.
Minutes from the break it should have been two when a fine run and pass from Kewell set up Riise on the left. His first shot was blocked by Molina and the follow-up was cleared in front of an open goal by Andrade.
Deportivo embarked on the second-half with more determination.
That was underlined when substitute Pablo Amo hurled himself at a free-kick on the far post which left Chris Kirkland flattened in the back of the net, although Biscan caught the 'keeper in the face.
Ireland international Steve Finnan replaced Kewell after 58 minutes and Garcia pushed forward but it was the running of Hamann and Biscan in the centre that was critical for Liverpool.
Andrade was booked for bringing down Baros on a rare break for the Merseysiders.
Eight minutes from time Liverpool wasted a chance to settle things when Baros put Garcia through but he allowed the ball run to Molina.
DEP LA CORUNA: Molina, Romero, Cesar (Pablo Amo 45), Andrade, Hector (Scaloni 65), Sergio, Valeron, Duscher, Victor, Pandiani (Tristan 59), Luque.
LIVERPOOL: Kirkland, Josemi, Hyypia, Carragher, Traore, Riise, Luis Garcia (Alonso 91), Hamann, Biscan, Kewell (Finnan 58), Baros (Sinama Pongolle 84).
Referee: W Stark (Germany).