Joint top of the group when defeat would have anchored them bottom, their prospects have soared.
A win in Monaco in three weeks' time will propel them into the knockout stage, and little here suggested they are incapable of winning there. This hugely encouraging victory was gleaned courtesy of an own-goal and confirmed to a joyous chorus of "Igor, Igor" hollered from the visiting fans, their adulation aimed at the usually maligned Igor Biscan.
Where the Croatian midfielder was once painted as a figure of fun, his dozy expression befitting his rather ponderous and occasional displays in the first team, Biscan is a player reborn these days.
Thrust into this contest after Xabi Alonso succumbed to a calf strain, Biscan was immense. His fine goal speared at Fulham last month had set an improved tone, and that has been maintained. Rugged tackling is now combined with genuine finesse. On his first European start since he was sent off at Marseille in March, he looked the £5m player Liverpool thought they were buying from Croatia Zagreb four years ago.
The magnificence of his battering run through the centre of Deportivo's disbelieving midfield after 14 minutes was matched by the weight of the pass which sent John-Arne Riise free down the left. The Norwegian's fizzed cross prompted panic in the six-yard box, the hapless Jorge Andrade deflecting the loose ball beyond José Molina as Milan Baros hovered menacingly.
That was the game's single decisive moment, though this whole performance could prove to be the defining moment of Biscan's Liverpool career. "Igor played very well alongside [Dietmar] Hamann, and it is encouraging to see players who have not been playing so much doing so well when they get in the team," said Benítez, whose style at Valencia revolved around using a squad system. "We had problems before the game because Xabi could only have played for 45 minutes, no more. But Igor played so well we didn't need to use him."
The £10.7m Basque emerged from the bench for a few moments in injury-time near the end, playing a small part in his first ever victory over a Deportivo side. Benítez, a two-times championship-winning manager, has nothing else to achieve in Spain, but this was still a sweet return. Alonso had joined Steven Gerrard and Djibril Cissé as absentees - some £50m worth of crocked talent - yet Liverpool never looked a side shorn of their main sources of inspiration.
Their incisive passes cut swaths through the Spanish team, who had proved impregnable at the Riazor in 10 of their previous 12 matches in this competition. That defensive surety had been eroded after only 27 seconds, Luis García lofting the ball over Andrade for Baros to gather only for the Czech to delay his shot and fail adequately to swerve away from the advancing Molina. The goalkeeper smothered the chance, but the locals never recovered from the shock.
Had Baros shown more awareness and Harry Kewell more confidence, Liverpool could have run riot. The Australian's self-doubt showed up three minutes from half-time, his sprint at the retreating centre-halves not culminating in a shot but in a veer left across the area. He fed Riise, who hammered his first shot at Molina, then the full-back saw his follow-up scrambled from the line by Andrade.
The Galicians, in response, appeared every bit the only side yet to score in the competition proper this season. The nearest they came was when Pablo Amo narrowly failed to convert Alberto Luque's whipped free-kick - the crowd showering their beleaguered coach Javier Irureta with abuse at the final whistle.
The home side's shortcomings must be acknowledged when putting this success into context, though confidence is sweeping through Liverpool's squad. A threadbare squad suddenly appears full of options, with Djimi Traoré a man transformed as he benefits from Jamie Carragher's assured presence at his side.
The Frenchman's astute challenge snuffed out Diego Tristán when he threatened late on, but they rarely looked likely to ship an equaliser. "That was the best Liverpool performance I have overseen," added Benítez. "Now our destiny is in our own hands."