Graeme Souness' side were beaten by goals in each half from Ruben Castro and Jorge Andrade but a stunning volley from Lee Bowyer limited the damage.
Deportivo provided a stern test for the injury-hit Magpies but despite their extra class and a number of chances, they were unable to put the tie beyond reach.
With an away goal to their name, Newcastle - who had been missing the likes of Scott Parker, Emre, Jermaine Jenas, Kieron Dyer, Shola Ameobi and Titus Bramble - still have a chance of an upset.
Newcastle were determined not to be intimidated and made their presence felt with some fierce challenges early on.
Stephen Carr and Nicky Butt - who recovered from a stomach bug to take his place in the side - both went in strongly for 50-50 balls.
James Milner also felt the full force of a challenge from Aldo Duscher and later required stitches for a cut near the eye.
But Deportivo soon began to assert themselves and looked the more composed team throughout the first half.
Their first chance came after just two minutes when the dangerous Pedro Munitis broke clear down the right and pulled the ball back towards goal but Jean-Alain Boumsong cleared.
Newcastle failed to heed the warning and Munitis exposed their weakness down the same flank to create the opening goal eight minutes later.
Steven Taylor was caught flat-footed as Munitis broke into the box and squared for Ruben to sidefoot home from six yards.
For a short spell it looked as if the Magpies might fold under the pressure as Ruben scuffed another shot in front of goal and Munitis fired straight at Shay Given.
Newcastle showed more composure as the half wore on but they were unable to convert their increased possession into opportunities. They were almost caught out again just before the interval as the ubiquitous Munitis found space outside the area and sent another shot narrowly wide.
Newcastle finally started to show some attacking intent as the second half began and were instantly rewarded as they won a corner and claimed a superb 48th-minute equaliser through Bowyer.
Stephen Carr's cross was cleared only as far as the edge of the area by Deportivo and the ball fell perfectly for Bowyer, who connected with a sweet left-foot volley that flew into the top corner.
The game became scrappy after that with referee Serge Gumienny frequently blowing for free-kicks but, stirred back into life by the equaliser, Deportivo lifted their game.
Newcastle were back on level terms for just 10 minutes as Andrade ran unchallenged into the box from the halfway line and curled a superb finish past Given.
The visitors then appeared to be hanging on as Ruben volleyed just wide with an acrobatic effort and Given needed to be alert to keep out a deflected Munitis cross.
The Newcastle wall then did its job as the Spaniards won two free-kicks outside the area but failed to test Given.
Munitis again caused a few flutters in the box as the clock ran down but, despite riding their luck on occasions, Newcastle held on for a result their determination deserved.
Deportivo La Coruña (4-4-2) Molina; Manuel Pablo, Andrade, Romero, Capdevila; Munitis, Duscher (Victor, 70), Sergio, Luque (Scaloni, 61); Ruben Castro, Tristan (Valeron, 55). Substitutes not used: Gomes, Munua, Juanma, Hector.
Newcastle United (4-5-1): Given; Carr, Boumsong, Taylor, Babayaro; Milner (Chopra, 71), Bowyer, Butt, Faye, N'Zogbia; Shearer. Substitutes not used: Elliott, Harper, Ramage, Brittain, Huntington.
Referee: S Gumienny (Bel).
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