Glenn Roeder's side only required one point, but followed up their solitary goal wins over Fenerbahce and at Palermo with this odd goal in three success.
It looked worrying for Newcastle when they trailed to a stunning early strike from Fabian Canobbia, but they went on to win courtesy of two headers to stretch their unbeaten cup run this season to 11 matches..
Antoine Sibierski, who headed the winner against Fenerbahce levelled it up by the break and then young central defender Steven Taylor was the matchwinner with a late downward header for his first senior goal.
United had a glorious opportunity to go ahead inside the opening 20 seconds. Nobby Solano found Sibierski down the right flank. The Frenchman laid the ball to James Milner who whipped in a deep cross for the unmarked Albert Luque, who pulled his shot across the goal and wide.
It was the slick-passing Celta who went in front after nine minutes with a delightful goal from Canobbio. United failed to clear a cross and Canobbio then brushed Emre off the ball before curling a perfect shot beyond the despairing Shay Given.
Buoyed by the goal, Celta then dominated the next half hour with United at times chasing shadows as Nene twice tested Given.
Newcastle needed a goal before the break and it came after 37 minutes when Milner got the better of Diego Placente and hoisted in a first-time cross that saw Sibierski climb above the Celta defence to head home.
It could have been better on the stroke of half-time when a Luque free-kick picked out Titus Bramble who headed wide.
There was a difference in United after the break and it was noticeable that they were quicker chasing Celta down when they were in possession.
Just after the hour mark, Celta flashed a warning from the most unlikely of sources in right-back Angel Lopez. The defender was allowed to work his way into a shooting position and fizzed a drive across the face of goal.
Substitute Gustavo Lopez then tried his luck with an audacious shot from the halfway line that went over the top with Given furiously back-pedalling.
But it was United who were to have the final say four minutes from time with Taylor's first goal for the first team.
Luque swung over a right-wing corner and Taylor found enough room to plant a firm downward header that resulted in a one-man celebration that stretched almost the length of the field.