David O'Leary's men defended from the front as strikers Mark Viduka and Smith muscled and bustled their way round the Spaniards.
Leeds took a huge step towards the semi-final of the European Cup as they humbled the Spanish Champions at Elland Road.
It was the first time for 26 years that the Yorskshire club had reached this far in a European competition.
And goals from Ian Harte, Alan Smith and Rio Ferdinand made sure it was a night to remember.
Leeds intimidated the Spanish champions into submission in an electrifying home display.
Far from being humbled by comments from the Deportivo La Coruna camp that Leeds were the competition's weakest link, the jibe spurred them into their best European performance.
David O'Leary's men defended from the front as strikers Mark Viduka and Smith muscled and bustled their way round the Spaniards.
It was Smith's determination not to relinquish the ball on the edge of the area that led to the opening goal.
Cesar conceeded a 24th minute free-kick when he manhandled Smith to the floor.
And Harry Kewell caused enough havoc in the wall to create an opening for Harte to rifle home his fourth Champions League goal of the season
Leeds never looked back as they tore up the Yorkshire turf.
Kewell, absent for many of Leeds' European games this season, was an ever present threat down the left.
In one sensational move in the opening quarter of an hour, he slid past one defender, jinked around another and was unlucky to slice the final shot narrowly over the bar.
And Frenchman Olivier Dacourt was rampant in midfield, snapping up any remotely loose ball and unleashing a series of sweeping passes.
Both players were involved in Leeds' second goal, which came only five minutes after the restart.
The score had the best of Leeds stamped all over it.
First Viduka hassled the ball from the Deportivo centre-half, allowing partner Smith to rattle a raking right-foot shot on goal.
Molina just managed to tip the effort wide. But Leeds won the resulting corner, Dacourt feeding Kewell who hit a first-time cross right onto the head of Smith.
The 21-year-old buried the ball for his seventh European strike.
Fittingly, it saw him draw level with Leeds great Peter Lorimer in the list of the club's leading scorers of European goals.
Lorimer scored in Leeds' last European Cup quarter-final outing - against Anderlecht in 1975.
Harte was unlucky not to get his second with a second-half free-kick which he struck even better than his first-half effort, but the curl of the ball took it fractionally wide.
Deportivo did not manufacture anything resembling a goalscoring chance until the 55th minute but Dutch striker Roy Makaay presented Nigel Martyn with an easy save.
But if the Spanish coach hoped that might signal a revival for his shellshocked players, he was wrong.
Instead, Leeds cranked up the pace even further before Ferdinand leapt to power home a third goal on the hour mark.
The England defender could not have chosen his moment better to score his first goal for the club, meeting Kewell's corner with a meaty header.
It sent an ecstatic Elland Road into delirious chants of "Three nil to the weakest link".
Martyn had one nasty moment when he spilled a free-kick. But Dominic Matteo, in for the injured Lucas Radebe, stepped in to clear up the trouble.
It was a fortunate break but if, as the saying goes, you make your own luck, then Leeds deserved it by the bucketload.